Theorem rexlimiva 2657

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 2656	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2205  ∃wrex 2523

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583  ax-i5r 1584

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-ral 2527  df-rex 2528

This theorem is referenced by:  unon  4635  reg2exmidlema  4658  ssfilem  7132  ssfilemd  7134  diffitest  7146  fival  7259  elfi2  7261  fi0  7264  djuss  7363  updjud  7375  enumct  7408  finnum  7481  dmaddpqlem  7694  nqpi  7695  nq0nn  7759  recexprlemm  7941  iswrd  11230  wrdf  11234  rexanuz  11677  r19.2uz  11682  maxleast  11902  fsum2dlemstep  12124  fisumcom2  12128  fprod2dlemstep  12312  fprodcom2fi  12316  0dvds  12501  even2n  12564  m1expe  12589  m1exp1  12591  modprm0  12956  gsumval2  13627  dfgrp2  13757  epttop  14972  neipsm  15036  tgioo  15436  sin0pilem2  15664  pilem3  15665  perfect  15886  clwwlkn1loopb  16432  bj-nn0suc  16751  bj-nn0sucALT  16765  trirec0xor  16846