vtoclegft

Description: Implicit substitution of a class for a setvar variable. (Closed theorem version of vtoclef .) (Contributed by NM, 7-Nov-2005) (Revised by Mario Carneiro, 11-Oct-2016)

		Ref	Expression
	Assertion	vtoclegft	$⊢ (A \in B \land Ⅎ x φ \land \forall x (x = A \to φ)) \to φ$

Proof

Step	Hyp	Ref	Expression
1		elisset	$⊢ A \in B \to \exists x x = A$
2		exim	$⊢ \forall x (x = A \to φ) \to (\exists x x = A \to \exists x φ)$
3	1 2	mpan9	$⊢ (A \in B \land \forall x (x = A \to φ)) \to \exists x φ$
4	3	3adant2	$⊢ (A \in B \land Ⅎ x φ \land \forall x (x = A \to φ)) \to \exists x φ$
5		19.9t	$⊢ Ⅎ x φ \to (\exists x φ \leftrightarrow φ)$
6	5	3ad2ant2	$⊢ (A \in B \land Ⅎ x φ \land \forall x (x = A \to φ)) \to (\exists x φ \leftrightarrow φ)$
7	4 6	mpbid	$⊢ (A \in B \land Ⅎ x φ \land \forall x (x = A \to φ)) \to φ$

Metamath Proof Explorer

Theorem vtoclegft

Proof