Theorem rexlimdvaa 2663

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 2662	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒))

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:  rexlimddv  2667  nnsucuniel  6741  omp1eomlem  7398  ctmlemr  7412  mulgt0sr  8109  axpre-suploclemres  8232  cnegex  8467  receuap  8960  recapb  8962  rexanuz  11698  climcaucn  12061  fsumiun  12188  dvdsval2  12501  nninfctlemfo  12761  prmind2  12842  pcprmpw2  13056  pockthg  13080  dvdsrvald  14323  dvdsrd  14324  dvdsrex  14328  unitgrp  14346  isnzr2  14414  znunit  14919  tgcl  15041  neiint  15122  restopnb  15158  iscnp4  15195  blssexps  15406  blssex  15407  lgsne0  16023  lgsquadlem1  16062