Theorem sbcied 3022

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 1539	. 2 |- F/ x ph
4		nfvd 1540	. 2 |- ( ph -> F/ x ch )
5	1, 2, 3, 4	sbciedf 3021	1 |- ( ph -> ( [. A / x ]. ps <-> ch ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   [.wsbc 2985

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-sbc 2986

This theorem is referenced by:  sbcied2  3023  sbc2iedv  3058  sbc3ie  3059  sbcralt  3062  sbcrext  3063  euotd  4283  riota5f  5898  issrg  13461  islmod  13787