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)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ∈ wcel 2209  ∃wrex 2529

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

This theorem is used by:  unon  4658  reg2exmidlema  4681  ssfilem  7177  ssfilemd  7179  diffitest  7191  fival  7304  elfi2  7306  fi0  7309  djuss  7411  updjud  7423  enumct  7456  finnum  7529  dmaddpqlem  7745  nqpi  7746  nq0nn  7810  recexprlemm  7992  iswrd  11322  wrdf  11326  rexanuz  11770  r19.2uz  11775  maxleast  11996  fsum2dlemstep  12220  fisumcom2  12224  fprod2dlemstep  12408  fprodcom2fi  12412  0dvds  12597  even2n  12660  m1expe  12685  m1exp1  12687  modprm0  13056  gzsumval2  13767  dfgrp2  13885  epttop  15282  neipsm  15346  tgioo  15746  sin0pilem2  15975  pilem3  15976  perfect  16262  clwwlkn1loopb  16827  bj-nn0suc  17156  bj-nn0sucALT  17170  trirec0xor  17261