Theorem spcimedv 2684

Description: Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypotheses

Ref	Expression
spcimdv.1	|- ( ph -> A e. B )
spcimedv.2	|- ( ( ph /\ x = A ) -> ( ch -> ps ) )

Assertion

Ref	Expression
spcimedv	|- ( ph -> ( ch -> E. x ps ) )

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

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

Proof of Theorem spcimedv

Step	Hyp	Ref	Expression
1		spcimedv.2	. . . 4 |- ( ( ph /\ x = A ) -> ( ch -> ps ) )
2	1	ex 113	. . 3 |- ( ph -> ( x = A -> ( ch -> ps ) ) )
3	2	alrimiv 1795	. 2 |- ( ph -> A. x ( x = A -> ( ch -> ps ) ) )
4		spcimdv.1	. 2 |- ( ph -> A e. B )
5		nfv 1461	. . 3 |- F/ x ch
6		nfcv 2219	. . 3 |- F/_ x A
7	5, 6	spcimegft 2676	. 2 |- ( A. x ( x = A -> ( ch -> ps ) ) -> ( A e. B -> ( ch -> E. x ps ) ) )
8	3, 4, 7	sylc 61	1 |- ( ph -> ( ch -> E. x ps ) )

Syntax hints:   -> wi 4   /\ wa 102   A.wal 1282   = wceq 1284   E.wex 1421   e. wcel 1433

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603

This theorem is referenced by:  rspcimedv  2703