Theorem rexlimdvaa 2652

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 2651	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒))

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2202  ∃wrex 2512

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 2516  df-rex 2517

This theorem is referenced by:  rexlimddv  2656  nnsucuniel  6706  omp1eomlem  7336  ctmlemr  7350  mulgt0sr  8041  axpre-suploclemres  8164  cnegex  8400  receuap  8892  recapb  8894  rexanuz  11609  climcaucn  11972  fsumiun  12099  dvdsval2  12412  nninfctlemfo  12672  prmind2  12753  pcprmpw2  12967  pockthg  12991  dvdsrvald  14169  dvdsrd  14170  dvdsrex  14174  unitgrp  14192  isnzr2  14260  znunit  14735  tgcl  14855  neiint  14936  restopnb  14972  iscnp4  15009  blssexps  15220  blssex  15221  lgsne0  15837  lgsquadlem1  15876