Theorem rexlimiva 2663

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 ∈ 𝐴 ∧ 𝜑) → 𝜓)

Assertion

Ref	Expression
rexlimiva	⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)

Distinct variable group:   𝜓,𝑥

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

Proof of Theorem rexlimiva

Step	Hyp	Ref	Expression
1		rexlimiva.1	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓)
2	1	ex 115	. 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))
3	2	rexlimiv 2662	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2209  ∃wrex 2529

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 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-i5r 1588

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-ral 2533  df-rex 2534

This theorem is referenced by:  unon  4656  reg2exmidlema  4679  ssfilem  7170  ssfilemd  7172  diffitest  7184  fival  7297  elfi2  7299  fi0  7302  djuss  7403  updjud  7415  enumct  7448  finnum  7521  dmaddpqlem  7737  nqpi  7738  nq0nn  7802  recexprlemm  7984  iswrd  11287  wrdf  11291  rexanuz  11735  r19.2uz  11740  maxleast  11960  fsum2dlemstep  12182  fisumcom2  12186  fprod2dlemstep  12370  fprodcom2fi  12374  0dvds  12559  even2n  12622  m1expe  12647  m1exp1  12649  modprm0  13014  gzsumval2  13694  dfgrp2  13812  epttop  15117  neipsm  15181  tgioo  15581  sin0pilem2  15809  pilem3  15810  perfect  16032  clwwlkn1loopb  16578  bj-nn0suc  16907  bj-nn0sucALT  16921  trirec0xor  17002