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  7410  updjud  7422  enumct  7455  finnum  7528  dmaddpqlem  7744  nqpi  7745  nq0nn  7809  recexprlemm  7991  iswrd  11320  wrdf  11324  rexanuz  11768  r19.2uz  11773  maxleast  11994  fsum2dlemstep  12217  fisumcom2  12221  fprod2dlemstep  12405  fprodcom2fi  12409  0dvds  12594  even2n  12657  m1expe  12682  m1exp1  12684  modprm0  13053  gzsumval2  13763  dfgrp2  13881  epttop  15240  neipsm  15304  tgioo  15704  sin0pilem2  15933  pilem3  15934  perfect  16199  clwwlkn1loopb  16759  bj-nn0suc  17088  bj-nn0sucALT  17102  trirec0xor  17192