Theorem spcimedv 2850

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 115	. . 3 |- ( ph -> ( x = A -> ( ch -> ps ) ) )
3	2	alrimiv 1888	. 2 |- ( ph -> A. x ( x = A -> ( ch -> ps ) ) )
4		spcimdv.1	. 2 |- ( ph -> A e. B )
5		nfv 1542	. . 3 |- F/ x ch
6		nfcv 2339	. . 3 |- F/_ x A
7	5, 6	spcimegft 2842	. 2 |- ( A. x ( x = A -> ( ch -> ps ) ) -> ( A e. B -> ( ch -> E. x ps ) ) )
8	3, 4, 7	sylc 62	1 |- ( ph -> ( ch -> E. x ps ) )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1362   = wceq 1364   E.wex 1506   e. wcel 2167

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765

This theorem is referenced by:  rspcimedv  2870  fihashf1rn  10880