Theorem sbcied 3069

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 1577	. 2 |- F/ x ph
4		nfvd 1578	. 2 |- ( ph -> F/ x ch )
5	1, 2, 3, 4	sbciedf 3068	1 |- ( ph -> ( [. A / x ]. ps <-> ch ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2202   [.wsbc 3032

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-sbc 3033

This theorem is referenced by:  sbcied2  3070  sbc2iedv  3105  sbc3ie  3106  sbcralt  3109  sbcrext  3110  euotd  4353  riota5f  6008  wrdind  11369  issrg  14059  islmod  14387