Theorem sbcied 2945

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 2944	1 |- ( ph -> ( [. A / x ]. ps <-> ch ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   [.wsbc 2909

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-sbc 2910

This theorem is referenced by:  sbcied2  2946  sbc2iedv  2981  sbc3ie  2982  sbcralt  2985  sbcrext  2986  euotd  4176  riota5f  5754