Theorem rexlimiva 2617

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 2616	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2175  ∃wrex 2484

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 1469  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-4 1532  ax-17 1548  ax-ial 1556  ax-i5r 1557

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-ral 2488  df-rex 2489

This theorem is referenced by:  unon  4558  reg2exmidlema  4581  ssfilem  6971  diffitest  6983  fival  7071  elfi2  7073  fi0  7076  djuss  7171  updjud  7183  enumct  7216  finnum  7289  dmaddpqlem  7489  nqpi  7490  nq0nn  7554  recexprlemm  7736  iswrd  10994  wrdf  10998  rexanuz  11270  r19.2uz  11275  maxleast  11495  fsum2dlemstep  11716  fisumcom2  11720  fprod2dlemstep  11904  fprodcom2fi  11908  0dvds  12093  even2n  12156  m1expe  12181  m1exp1  12183  modprm0  12548  gsumval2  13200  dfgrp2  13330  epttop  14533  neipsm  14597  tgioo  14997  sin0pilem2  15225  pilem3  15226  perfect  15444  bj-nn0suc  15862  bj-nn0sucALT  15876  trirec0xor  15946