Theorem sbcied 2973

Description: Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.)

Hypotheses

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

Assertion

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

Distinct variable groups:   x, A   ph, x   ch, x

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

Proof of Theorem sbcied

Step	Hyp	Ref	Expression
1		sbcied.1	. 2 |- ( ph -> A e. V )
2		sbcied.2	. 2 |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )
3		nfv 1508	. 2 |- F/ x ph
4		nfvd 1509	. 2 |- ( ph -> F/ x ch )
5	1, 2, 3, 4	sbciedf 2972	1 |- ( ph -> ( [. A / x ]. ps <-> ch ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1335   e. wcel 2128   [.wsbc 2937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-sbc 2938

This theorem is referenced by:  sbcied2  2974  sbc2iedv  3009  sbc3ie  3010  sbcralt  3013  sbcrext  3014  euotd  4214  riota5f  5804