Theorem sbciedf 2875

Description: Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 29-Dec-2014.)

Hypotheses

Ref	Expression
sbcied.1	|- ( ph -> A e. V )
sbcied.2	|- ( ( ph /\ x = A ) -> ( ps <-> ch ) )
sbciedf.3	|- F/ x ph
sbciedf.4	|- ( ph -> F/ x ch )

Assertion

Ref	Expression
sbciedf	|- ( ph -> ( [. A / x ]. ps <-> ch ) )

Distinct variable group:   x, A

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

Proof of Theorem sbciedf

Step	Hyp	Ref	Expression
1		sbcied.1	. 2 |- ( ph -> A e. V )
2		sbciedf.4	. 2 |- ( ph -> F/ x ch )
3		sbciedf.3	. . 3 |- F/ x ph
4		sbcied.2	. . . 4 |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )
5	4	ex 114	. . 3 |- ( ph -> ( x = A -> ( ps <-> ch ) ) )
6	3, 5	alrimi 1461	. 2 |- ( ph -> A. x ( x = A -> ( ps <-> ch ) ) )
7		sbciegft 2870	. 2 |- ( ( A e. V /\ F/ x ch /\ A. x ( x = A -> ( ps <-> ch ) ) ) -> ( [. A / x ]. ps <-> ch ) )
8	1, 2, 6, 7	syl3anc 1175	1 |- ( ph -> ( [. A / x ]. ps <-> ch ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> 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:  sbcied  2876  sbc2iegf  2910  csbiebt  2968  sbcnestgf  2980  ovmpt2dxf  5784