ssralv2VD

Description: Quantification restricted to a subclass for two quantifiers. ssralv for two quantifiers. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. ssralv2 is ssralv2VD without virtual deductions and was automatically derived from ssralv2VD .

		Ref	Expression
	Assertion	ssralv2VD	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷 ) → ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐷 𝜑 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐶 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		ax-5	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷 ) → ∀ 𝑥 ( 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷 ) )
2		hbra1	⊢ ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐷 𝜑 → ∀ 𝑥 ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐷 𝜑 )
3		idn1	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷 ) ▶ ( 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷 ) )
4		simpr	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷 ) → 𝐶 ⊆ 𝐷 )
5	3 4	e1a	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷 ) ▶ 𝐶 ⊆ 𝐷 )
6		idn3	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷 ) , ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐷 𝜑 , 𝑥 ∈ 𝐴 ▶ 𝑥 ∈ 𝐴 )
7		simpl	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷 ) → 𝐴 ⊆ 𝐵 )
8	3 7	e1a	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷 ) ▶ 𝐴 ⊆ 𝐵 )
9		idn2	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷 ) , ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐷 𝜑 ▶ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐷 𝜑 )
10		ssralv	⊢ ( 𝐴 ⊆ 𝐵 → ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐷 𝜑 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐷 𝜑 ) )
11	8 9 10	e12	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷 ) , ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐷 𝜑 ▶ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐷 𝜑 )
12		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐷 𝜑 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ∈ 𝐷 𝜑 ) )
13	12	biimpi	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐷 𝜑 → ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ∈ 𝐷 𝜑 ) )
14	11 13	e2	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷 ) , ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐷 𝜑 ▶ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ∈ 𝐷 𝜑 ) )
15		sp	⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ∈ 𝐷 𝜑 ) → ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ∈ 𝐷 𝜑 ) )
16	14 15	e2	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷 ) , ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐷 𝜑 ▶ ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ∈ 𝐷 𝜑 ) )
17		pm2.27	⊢ ( 𝑥 ∈ 𝐴 → ( ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ∈ 𝐷 𝜑 ) → ∀ 𝑦 ∈ 𝐷 𝜑 ) )
18	6 16 17	e32	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷 ) , ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐷 𝜑 , 𝑥 ∈ 𝐴 ▶ ∀ 𝑦 ∈ 𝐷 𝜑 )
19		ssralv	⊢ ( 𝐶 ⊆ 𝐷 → ( ∀ 𝑦 ∈ 𝐷 𝜑 → ∀ 𝑦 ∈ 𝐶 𝜑 ) )
20	5 18 19	e13	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷 ) , ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐷 𝜑 , 𝑥 ∈ 𝐴 ▶ ∀ 𝑦 ∈ 𝐶 𝜑 )
21	20	in3	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷 ) , ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐷 𝜑 ▶ ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ∈ 𝐶 𝜑 ) )
22	1 2 21	gen21nv	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷 ) , ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐷 𝜑 ▶ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ∈ 𝐶 𝜑 ) )
23		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐶 𝜑 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ∈ 𝐶 𝜑 ) )
24	23	biimpri	⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ∈ 𝐶 𝜑 ) → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐶 𝜑 )
25	22 24	e2	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷 ) , ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐷 𝜑 ▶ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐶 𝜑 )
26	25	in2	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷 ) ▶ ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐷 𝜑 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐶 𝜑 ) )
27	26	in1	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷 ) → ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐷 𝜑 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐶 𝜑 ) )

1::	\|- (. ( A C_ B /\ C C_ D ) ->. ( A C_ B /\ C C_ D ) ).
2::	\|- (. ( A C_ B /\ C C_ D ) ,. A. x e. B A. y e. D ph ->. A. x e. B A. y e. D ph ).
3:1:	\|- (. ( A C_ B /\ C C_ D ) ->. A C_ B ).
4:3,2:	\|- (. ( A C_ B /\ C C_ D ) ,. A. x e. B A. y e. D ph ->. A. x e. A A. y e. D ph ).
5:4:	\|- (. ( A C_ B /\ C C_ D ) ,. A. x e. B A. y e. D ph ->. A. x ( x e. A -> A. y e. D ph ) ).
6:5:	\|- (. ( A C_ B /\ C C_ D ) ,. A. x e. B A. y e. D ph ->. ( x e. A -> A. y e. D ph ) ).
7::	\|- (. ( A C_ B /\ C C_ D ) ,. A. x e. B A. y e. D ph , x e. A ->. x e. A ).
8:7,6:	\|- (. ( A C_ B /\ C C_ D ) ,. A. x e. B A. y e. D ph , x e. A ->. A. y e. D ph ).
9:1:	\|- (. ( A C_ B /\ C C_ D ) ->. C C_ D ).
10:9,8:	\|- (. ( A C_ B /\ C C_ D ) ,. A. x e. B A. y e. D ph , x e. A ->. A. y e. C ph ).
11:10:	\|- (. ( A C_ B /\ C C_ D ) ,. A. x e. B A. y e. D ph ->. ( x e. A -> A. y e. C ph ) ).
12::	\|- ( ( A C_ B /\ C C_ D ) -> A. x ( A C_ B /\ C C_ D ) )
13::	\|- ( A. x e. B A. y e. D ph -> A. x A. x e. B A. y e. D ph )
14:12,13,11:	\|- (. ( A C_ B /\ C C_ D ) ,. A. x e. B A. y e. D ph ->. A. x ( x e. A -> A. y e. C ph ) ).
15:14:	\|- (. ( A C_ B /\ C C_ D ) ,. A. x e. B A. y e. D ph ->. A. x e. A A. y e. C ph ).
16:15:	\|- (. ( A C_ B /\ C C_ D ) ->. ( A. x e. B A. y e. D ph -> A. x e. A A. y e. C ph ) ).
qed:16:	\|- ( ( A C_ B /\ C C_ D ) -> ( A. x e. B A. y e. D ph -> A. x e. A A. y e. C ph ) )

Metamath Proof Explorer

Theorem ssralv2VD

Proof