Theorem sbciegf 2871

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 1384	. 2 |- A. x ( x = A -> ( ph <-> ps ) )
4		sbciegft 2870	. 2 |- ( ( A e. V /\ F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) ) -> ( [. A / x ]. ph <-> ps ) )
5	1, 3, 4	mp3an23 1266	1 |- ( A e. V -> ( [. A / x ]. ph <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1288   = wceq 1290   F/wnf 1395   e. wcel 1439   [.wsbc 2841

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-sbc 2842

This theorem is referenced by:  sbcieg  2872  iunxsngf  3813  opelopabf  4110  eqerlem  6337