Theorem spc2ev 3152

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 3146	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( ps -> E. x E. y ph ) )
5	1, 2, 4	mp2an 670	1 |- ( ps -> E. x E. y ph )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   = wceq 1405   E.wex 1633   e. wcel 1842   _Vcvv 3059

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-ext 2380

This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-v 3061

This theorem is referenced by:  relop  4974  endisj  7642  dcomex  8859  axcnre  9571  constr3cyclpe  25080  3v3e3cycl2  25081  qqhval2  28415  itg2addnclem3  31441  wlkc  37979