Theorem sbciegf 2940

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 1425	. 2 |- A. x ( x = A -> ( ph <-> ps ) )
4		sbciegft 2939	. 2 |- ( ( A e. V /\ F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) ) -> ( [. A / x ]. ph <-> ps ) )
5	1, 3, 4	mp3an23 1307	1 |- ( A e. V -> ( [. A / x ]. ph <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1329   = wceq 1331   F/wnf 1436   e. wcel 1480   [.wsbc 2909

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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-sbc 2910

This theorem is referenced by:  sbcieg  2941  iunxsngf  3890  opelopabf  4196  eqerlem  6460