Theorem rexlimiva 2473

Description: Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 18-Dec-2006.)

Hypothesis

Ref	Expression
rexlimiva.1	|- ( ( x e. A /\ ph ) -> ps )

Assertion

Ref	Expression
rexlimiva	|- ( E. x e. A ph -> ps )

Distinct variable group:   ps, x

Allowed substitution hints:   ph( x)   A( x)

Proof of Theorem rexlimiva

Step	Hyp	Ref	Expression
1		rexlimiva.1	. . 3 |- ( ( x e. A /\ ph ) -> ps )
2	1	ex 113	. 2 |- ( x e. A -> ( ph -> ps ) )
3	2	rexlimiv 2472	1 |- ( E. x e. A ph -> ps )

Syntax hints:   -> wi 4   /\ wa 102   e. wcel 1434   E.wrex 2350

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468  ax-i5r 1469

This theorem depends on definitions:  df-bi 115  df-nf 1391  df-ral 2354  df-rex 2355

This theorem is referenced by:  unon  4257  reg2exmidlema  4279  ssfilem  6400  diffitest  6411  finnum  6501  dmaddpqlem  6618  nqpi  6619  nq0nn  6683  recexprlemm  6865  rexanuz  10001  r19.2uz  10006  maxleast  10226  0dvds  10349  even2n  10407  m1expe  10432  m1exp1  10434  bj-nn0suc  10902  bj-nn0sucALT  10916