bnj1133

Description: Technical lemma for bnj69 . This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Hypotheses	bnj1133.3	$⊢ D = ω ∖ \{\emptyset\}$
	bnj1133.5	$⊢ χ \leftrightarrow (n \in D \land f Fn n \land φ \land ψ)$
	bnj1133.7	$⊢ τ \leftrightarrow \forall j \in n (j E i \to \underset{˙}{[} j / i \underset{˙}{]} θ)$
	bnj1133.8	$⊢ (i \in n \land τ) \to θ$
Assertion	bnj1133	$⊢ χ \to \forall i \in n θ$

Proof

Step	Hyp	Ref	Expression
1		bnj1133.3	$⊢ D = ω ∖ \{\emptyset\}$
2		bnj1133.5	$⊢ χ \leftrightarrow (n \in D \land f Fn n \land φ \land ψ)$
3		bnj1133.7	$⊢ τ \leftrightarrow \forall j \in n (j E i \to \underset{˙}{[} j / i \underset{˙}{]} θ)$
4		bnj1133.8	$⊢ (i \in n \land τ) \to θ$
5	1	bnj1071	$⊢ n \in D \to E Fr n$
6	2 5	bnj769	$⊢ χ \to E Fr n$
7		impexp	$⊢ ((i \in n \land τ) \to θ) \leftrightarrow (i \in n \to (τ \to θ))$
8	7	bicomi	$⊢ (i \in n \to (τ \to θ)) \leftrightarrow ((i \in n \land τ) \to θ)$
9	8	albii	$⊢ \forall i (i \in n \to (τ \to θ)) \leftrightarrow \forall i ((i \in n \land τ) \to θ)$
10	9 4	mpgbir	$⊢ \forall i (i \in n \to (τ \to θ))$
11		df-ral	$⊢ \forall i \in n (τ \to θ) \leftrightarrow \forall i (i \in n \to (τ \to θ))$
12	10 11	mpbir	$⊢ \forall i \in n (τ \to θ)$
13		vex	$⊢ n \in V$
14	13 3	bnj110	$⊢ (E Fr n \land \forall i \in n (τ \to θ)) \to \forall i \in n θ$
15	6 12 14	sylancl	$⊢ χ \to \forall i \in n θ$

Metamath Proof Explorer

Theorem bnj1133

Proof