Theorem rexlimiva 2428

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 108	. 2 |- ( x e. A -> ( ph -> ps ) )
3	2	rexlimiv 2427	1 |- ( E. x e. A ph -> ps )

Syntax hints:   -> wi 4   /\ wa 97   e. wcel 1393   E.wrex 2307

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-ral 2311  df-rex 2312

This theorem is referenced by:  unon  4237  reg2exmidlema  4259  ssfiexmid  6336  diffitest  6344  finnum  6363  dmaddpqlem  6475  nqpi  6476  nq0nn  6540  recexprlemm  6722  rexanuz  9587  r19.2uz  9591  bj-nn0suc  10089  bj-nn0sucALT  10103