Theorem spcimedv 2995

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		spcimdv.1	. . . 4 |- ( ph -> A e. B )
2		spcimedv.2	. . . . 5 |- ( ( ph /\ x = A ) -> ( ch -> ps ) )
3	2	con3d 127	. . . 4 |- ( ( ph /\ x = A ) -> ( -. ps -> -. ch ) )
4	1, 3	spcimdv 2993	. . 3 |- ( ph -> ( A. x -. ps -> -. ch ) )
5	4	con2d 109	. 2 |- ( ph -> ( ch -> -. A. x -. ps ) )
6		df-ex 1548	. 2 |- ( E. x ps <-> -. A. x -. ps )
7	5, 6	syl6ibr 219	1 |- ( ph -> ( ch -> E. x ps ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 359   A.wal 1546   E.wex 1547   = wceq 1649   e. wcel 1721

This theorem is referenced by:  hashf1rn  11591

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385

This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-v 2918