Theorem sbciedf 3064

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 1568	. 2 |- ( ph -> A. x ( x = A -> ( ps <-> ch ) ) )
7		sbciegft 3059	. 2 |- ( ( A e. V /\ F/ x ch /\ A. x ( x = A -> ( ps <-> ch ) ) ) -> ( [. A / x ]. ps <-> ch ) )
8	1, 2, 6, 7	syl3anc 1271	1 |- ( ph -> ( [. A / x ]. ps <-> ch ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1393   = wceq 1395   F/wnf 1506   e. wcel 2200   [.wsbc 3028

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-sbc 3029

This theorem is referenced by:  sbcied  3065  sbc2iegf  3099  csbiebt  3164  sbcnestgf  3176  ovmpodxf  6130