Theorem vtoclegft 2798

Description: Implicit substitution of a class for a setvar variable. (Closed theorem version of vtoclef 2799.) (Contributed by NM, 7-Nov-2005.) (Revised by Mario Carneiro, 11-Oct-2016.)

Assertion

Ref	Expression
vtoclegft	|- ( ( A e. B /\ F/ x ph /\ A. x ( x = A -> ph ) ) -> ph )

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   B( x)

Proof of Theorem vtoclegft

Step	Hyp	Ref	Expression
1		elisset 2740	. . . 4 |- ( A e. B -> E. x x = A )
2		exim 1587	. . . 4 |- ( A. x ( x = A -> ph ) -> ( E. x x = A -> E. x ph ) )
3	1, 2	mpan9 279	. . 3 |- ( ( A e. B /\ A. x ( x = A -> ph ) ) -> E. x ph )
4	3	3adant2 1006	. 2 |- ( ( A e. B /\ F/ x ph /\ A. x ( x = A -> ph ) ) -> E. x ph )
5		19.9t 1630	. . 3 |- ( F/ x ph -> ( E. x ph <-> ph ) )
6	5	3ad2ant2 1009	. 2 |- ( ( A e. B /\ F/ x ph /\ A. x ( x = A -> ph ) ) -> ( E. x ph <-> ph ) )
7	4, 6	mpbid 146	1 |- ( ( A e. B /\ F/ x ph /\ A. x ( x = A -> ph ) ) -> ph )

Syntax hints:   -> wi 4   <-> wb 104   /\ w3a 968   A.wal 1341   = wceq 1343   F/wnf 1448   E.wex 1480   e. wcel 2136

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-v 2728

This theorem is referenced by: (None)