Theorem vtoclegft 2875

Description: Implicit substitution of a class for a setvar variable. (Closed theorem version of vtoclef 2876.) (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 2814	. . . 4 |- ( A e. B -> E. x x = A )
2		exim 1645	. . . 4 |- ( A. x ( x = A -> ph ) -> ( E. x x = A -> E. x ph ) )
3	1, 2	mpan9 281	. . 3 |- ( ( A e. B /\ A. x ( x = A -> ph ) ) -> E. x ph )
4	3	3adant2 1040	. 2 |- ( ( A e. B /\ F/ x ph /\ A. x ( x = A -> ph ) ) -> E. x ph )
5		19.9t 1688	. . 3 |- ( F/ x ph -> ( E. x ph <-> ph ) )
6	5	3ad2ant2 1043	. 2 |- ( ( A e. B /\ F/ x ph /\ A. x ( x = A -> ph ) ) -> ( E. x ph <-> ph ) )
7	4, 6	mpbid 147	1 |- ( ( A e. B /\ F/ x ph /\ A. x ( x = A -> ph ) ) -> ph )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 1002   A.wal 1393   = wceq 1395   F/wnf 1506   E.wex 1538   e. wcel 2200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-v 2801

This theorem is referenced by: (None)