Theorem vtoclegftOLD

Description: Obsolete version of vtoclegft as of 26-Jan-2025. (Contributed by NM, 7-Nov-2005) (Revised by Mario Carneiro, 11-Oct-2016) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	vtoclegftOLD	$$⊢\left(A\in B\wedge Ⅎx\phi \wedge \forall x\left(x=A\to \phi \right)\right)\to \phi$$

Proof

Step	Hyp	Ref	Expression
1		elisset	$$⊢A\in B\to \exists xx=A$$
2		exim	$$⊢\forall x\left(x=A\to \phi \right)\to \left(\exists xx=A\to \exists x\phi \right)$$
3	1 2	mpan9	$$⊢\left(A\in B\wedge \forall x\left(x=A\to \phi \right)\right)\to \exists x\phi$$
4	3	3adant2	$$⊢\left(A\in B\wedge Ⅎx\phi \wedge \forall x\left(x=A\to \phi \right)\right)\to \exists x\phi$$
5		19.9t	$$⊢Ⅎx\phi \to \left(\exists x\phi \leftrightarrow \phi \right)$$
6	5	3ad2ant2	$$⊢\left(A\in B\wedge Ⅎx\phi \wedge \forall x\left(x=A\to \phi \right)\right)\to \left(\exists x\phi \leftrightarrow \phi \right)$$
7	4 6	mpbid	$$⊢\left(A\in B\wedge Ⅎx\phi \wedge \forall x\left(x=A\to \phi \right)\right)\to \phi$$