Theorem sbciegf 2794

Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)

Hypotheses

Ref	Expression
sbciegf.1	|- F/ x ps
sbciegf.2	|- ( x = A -> ( ph <-> ps ) )

Assertion

Ref	Expression
sbciegf	|- ( A e. V -> ( [. A / x ]. ph <-> ps ) )

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   ps( x)   V( x)

Proof of Theorem sbciegf

Step	Hyp	Ref	Expression
1		sbciegf.1	. 2 |- F/ x ps
2		sbciegf.2	. . 3 |- ( x = A -> ( ph <-> ps ) )
3	2	ax-gen 1338	. 2 |- A. x ( x = A -> ( ph <-> ps ) )
4		sbciegft 2793	. 2 |- ( ( A e. V /\ F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) ) -> ( [. A / x ]. ph <-> ps ) )
5	1, 3, 4	mp3an23 1224	1 |- ( A e. V -> ( [. A / x ]. ph <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 98   A.wal 1241   = wceq 1243   F/wnf 1349   e. wcel 1393   [.wsbc 2764

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765

This theorem is referenced by:  sbcieg  2795  opelopabf  4011  eqerlem  6137