Theorem spc2ev 3163

Description: Existential specialization, using implicit substitution. (Contributed by NM, 3-Aug-1995.)

Hypotheses

Ref	Expression
spc2ev.1	|- A e. _V
spc2ev.2	|- B e. _V
spc2ev.3	|- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )

Assertion

Ref	Expression
spc2ev	|- ( ps -> E. x E. y ph )

Distinct variable groups:   x, y, A   x, B, y   ps, x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem spc2ev

Step	Hyp	Ref	Expression
1		spc2ev.1	. 2 |- A e. _V
2		spc2ev.2	. 2 |- B e. _V
3		spc2ev.3	. . 3 |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
4	3	spc2egv 3157	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( ps -> E. x E. y ph ) )
5	1, 2, 4	mp2an 672	1 |- ( ps -> E. x E. y ph )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1370   E.wex 1587   e. wcel 1758   _Vcvv 3070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-ext 2430

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-v 3072

This theorem is referenced by:  relop  5090  th3qlem2  7309  endisj  7500  dcomex  8719  axcnre  9434  constr3cyclpe  23686  3v3e3cycl2  23687  qqhval2  26547  itg2addnclem3  28585