Theorem spc2ev 2860

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 2854	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( ps -> E. x E. y ph ) )
5	1, 2, 4	mp2an 426	1 |- ( ps -> E. x E. y ph )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   E.wex 1506   e. wcel 2167   _Vcvv 2763

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-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-ext 2178

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

This theorem is referenced by:  relop  4816  th3qlem2  6697  endisj  6883  axcnre  7946