cbv2

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . See cbv2w with disjoint variable conditions, not depending on ax-13 . (Contributed by NM, 5-Aug-1993) (Revised by Mario Carneiro, 3-Oct-2016) Format hypotheses to common style, avoid ax-10 . (Revised by Wolf Lammen, 10-Sep-2023) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cbv2.1	⊢ Ⅎ 𝑥 𝜑
	cbv2.2	⊢ Ⅎ 𝑦 𝜑
	cbv2.3	⊢ ( 𝜑 → Ⅎ 𝑦 𝜓 )
	cbv2.4	⊢ ( 𝜑 → Ⅎ 𝑥 𝜒 )
	cbv2.5	⊢ ( 𝜑 → ( 𝑥 = 𝑦 → ( 𝜓 ↔ 𝜒 ) ) )
Assertion	cbv2	⊢ ( 𝜑 → ( ∀ 𝑥 𝜓 ↔ ∀ 𝑦 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		cbv2.1	⊢ Ⅎ 𝑥 𝜑
2		cbv2.2	⊢ Ⅎ 𝑦 𝜑
3		cbv2.3	⊢ ( 𝜑 → Ⅎ 𝑦 𝜓 )
4		cbv2.4	⊢ ( 𝜑 → Ⅎ 𝑥 𝜒 )
5		cbv2.5	⊢ ( 𝜑 → ( 𝑥 = 𝑦 → ( 𝜓 ↔ 𝜒 ) ) )
6		biimp	⊢ ( ( 𝜓 ↔ 𝜒 ) → ( 𝜓 → 𝜒 ) )
7	5 6	syl6	⊢ ( 𝜑 → ( 𝑥 = 𝑦 → ( 𝜓 → 𝜒 ) ) )
8	1 2 3 4 7	cbv1	⊢ ( 𝜑 → ( ∀ 𝑥 𝜓 → ∀ 𝑦 𝜒 ) )
9		equcomi	⊢ ( 𝑦 = 𝑥 → 𝑥 = 𝑦 )
10		biimpr	⊢ ( ( 𝜓 ↔ 𝜒 ) → ( 𝜒 → 𝜓 ) )
11	9 5 10	syl56	⊢ ( 𝜑 → ( 𝑦 = 𝑥 → ( 𝜒 → 𝜓 ) ) )
12	2 1 4 3 11	cbv1	⊢ ( 𝜑 → ( ∀ 𝑦 𝜒 → ∀ 𝑥 𝜓 ) )
13	8 12	impbid	⊢ ( 𝜑 → ( ∀ 𝑥 𝜓 ↔ ∀ 𝑦 𝜒 ) )

Metamath Proof Explorer

Theorem cbv2

Proof