Theorem vtoclegft 2670

Description: Implicit substitution of a class for a setvar variable. (Closed theorem version of vtoclef 2671.) (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 2613	. . . 4 |- ( A e. B -> E. x x = A )
2		exim 1530	. . . 4 |- ( A. x ( x = A -> ph ) -> ( E. x x = A -> E. x ph ) )
3	1, 2	mpan9 275	. . 3 |- ( ( A e. B /\ A. x ( x = A -> ph ) ) -> E. x ph )
4	3	3adant2 957	. 2 |- ( ( A e. B /\ F/ x ph /\ A. x ( x = A -> ph ) ) -> E. x ph )
5		19.9t 1573	. . 3 |- ( F/ x ph -> ( E. x ph <-> ph ) )
6	5	3ad2ant2 960	. 2 |- ( ( A e. B /\ F/ x ph /\ A. x ( x = A -> ph ) ) -> ( E. x ph <-> ph ) )
7	4, 6	mpbid 145	1 |- ( ( A e. B /\ F/ x ph /\ A. x ( x = A -> ph ) ) -> ph )

Syntax hints:   -> wi 4   <-> wb 103   /\ w3a 919   A.wal 1282   = wceq 1284   F/wnf 1389   E.wex 1421   e. wcel 1433

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-v 2603

This theorem is referenced by: (None)