Theorem spcimedv 3193

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 133	. . . 4 |- ( ( ph /\ x = A ) -> ( -. ps -> -. ch ) )
4	1, 3	spcimdv 3191	. . 3 |- ( ph -> ( A. x -. ps -> -. ch ) )
5	4	con2d 115	. 2 |- ( ph -> ( ch -> -. A. x -. ps ) )
6		df-ex 1614	. 2 |- ( E. x ps <-> -. A. x -. ps )
7	5, 6	syl6ibr 227	1 |- ( ph -> ( ch -> E. x ps ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   A.wal 1393   = wceq 1395   E.wex 1613   e. wcel 1819

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111

This theorem is referenced by:  hashf1rn  12428  cshwsexa  12804  wwlktovfo  12908  uvcendim  19009  wlkiswwlk2  24824  wlknwwlknsur  24839  wlkiswwlksur  24846  wwlkextsur  24858  clwlkisclwwlklem2  24913  clwlksizeeq  24979