Theorem sbciedf 3000

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 115	. . 3 |- ( ph -> ( x = A -> ( ps <-> ch ) ) )
6	3, 5	alrimi 1522	. 2 |- ( ph -> A. x ( x = A -> ( ps <-> ch ) ) )
7		sbciegft 2995	. 2 |- ( ( A e. V /\ F/ x ch /\ A. x ( x = A -> ( ps <-> ch ) ) ) -> ( [. A / x ]. ps <-> ch ) )
8	1, 2, 6, 7	syl3anc 1238	1 |- ( ph -> ( [. A / x ]. ps <-> ch ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1351   = wceq 1353   F/wnf 1460   e. wcel 2148   [.wsbc 2964

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-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-sbc 2965

This theorem is referenced by:  sbcied  3001  sbc2iegf  3035  csbiebt  3098  sbcnestgf  3110  ovmpodxf  6003