Theorem rexlimiva 2620

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 2619	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2178  ∃wrex 2487

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 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-4 1534  ax-17 1550  ax-ial 1558  ax-i5r 1559

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-ral 2491  df-rex 2492

This theorem is referenced by:  unon  4577  reg2exmidlema  4600  ssfilem  6998  diffitest  7010  fival  7098  elfi2  7100  fi0  7103  djuss  7198  updjud  7210  enumct  7243  finnum  7316  dmaddpqlem  7525  nqpi  7526  nq0nn  7590  recexprlemm  7772  iswrd  11033  wrdf  11037  rexanuz  11414  r19.2uz  11419  maxleast  11639  fsum2dlemstep  11860  fisumcom2  11864  fprod2dlemstep  12048  fprodcom2fi  12052  0dvds  12237  even2n  12300  m1expe  12325  m1exp1  12327  modprm0  12692  gsumval2  13344  dfgrp2  13474  epttop  14677  neipsm  14741  tgioo  15141  sin0pilem2  15369  pilem3  15370  perfect  15588  bj-nn0suc  16099  bj-nn0sucALT  16113  trirec0xor  16186