Theorem rexlimdvaa 2669

Description: Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by Mario Carneiro, 15-Jun-2016.)

Hypothesis

Ref	Expression
rexlimdvaa.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) → 𝜒)

Assertion

Ref	Expression
rexlimdvaa	⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒))

Distinct variable groups:   𝜑,𝑥   𝜒,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem rexlimdvaa

Step	Hyp	Ref	Expression
1		rexlimdvaa.1	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) → 𝜒)
2	1	expr 375	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒))
3	2	rexlimdva 2668	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒))

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2209  ∃wrex 2529

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 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-i5r 1588

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

This theorem is referenced by:  rexlimddv  2673  nnsucuniel  6762  omp1eomlem  7428  ctmlemr  7442  mulgt0sr  8139  axpre-suploclemres  8262  cnegex  8498  receuap  8993  recapb  8995  rexanuz  11737  climcaucn  12100  fsumiun  12227  dvdsval2  12540  nninfctlemfo  12800  prmind2  12881  pcprmpw2  13095  pockthg  13119  dvdsrvald  14383  dvdsrd  14384  dvdsrex  14388  unitgrp  14406  isnzr2  14474  znunit  14977  tgcl  15148  neiint  15229  restopnb  15265  iscnp4  15302  blssexps  15513  blssex  15514  lgsne0  16140  lgsquadlem1  16179