Theorem sbciedf 3041

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 1546	. 2 |- ( ph -> A. x ( x = A -> ( ps <-> ch ) ) )
7		sbciegft 3036	. 2 |- ( ( A e. V /\ F/ x ch /\ A. x ( x = A -> ( ps <-> ch ) ) ) -> ( [. A / x ]. ps <-> ch ) )
8	1, 2, 6, 7	syl3anc 1250	1 |- ( ph -> ( [. A / x ]. ps <-> ch ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1371   = wceq 1373   F/wnf 1484   e. wcel 2178   [.wsbc 3005

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-sbc 3006

This theorem is referenced by:  sbcied  3042  sbc2iegf  3076  csbiebt  3141  sbcnestgf  3153  ovmpodxf  6094