Theorem rexlimdvaa 2584

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 373	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒))
3	2	rexlimdva 2583	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒))

Syntax hints:   → wi 4   ∧ wa 103   ∈ wcel 2136  ∃wrex 2445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-ral 2449  df-rex 2450

This theorem is referenced by:  rexlimddv  2588  nnsucuniel  6463  omp1eomlem  7059  ctmlemr  7073  mulgt0sr  7719  axpre-suploclemres  7842  cnegex  8076  receuap  8566  rexanuz  10930  climcaucn  11292  fsumiun  11418  dvdsval2  11730  prmind2  12052  pcprmpw2  12264  pockthg  12287  tgcl  12714  neiint  12795  restopnb  12831  iscnp4  12868  blssexps  13079  blssex  13080  lgsne0  13589