Theorem ralrimdva 2612

Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 2-Feb-2008.)

Hypothesis

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

Assertion

Ref	Expression
ralrimdva	⊢ (𝜑 → (𝜓 → ∀𝑥 ∈ 𝐴 𝜒))

Distinct variable groups:   𝜑,𝑥   𝜓,𝑥

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

Proof of Theorem ralrimdva

Step	Hyp	Ref	Expression
1		ralrimdva.1	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒))
2	1	ex 115	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒)))
3	2	com23 78	. 2 ⊢ (𝜑 → (𝜓 → (𝑥 ∈ 𝐴 → 𝜒)))
4	3	ralrimdv 2611	1 ⊢ (𝜑 → (𝜓 → ∀𝑥 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2202  ∀wral 2510

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 1495  ax-gen 1497  ax-4 1558  ax-17 1574

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-ral 2515

This theorem is referenced by:  ralxfrd  4559  isoselem  5961  isosolem  5965  findcard  7077  nnsub  9182  supinfneg  9829  infsupneg  9830  ublbneg  9847  expnlbnd2  10927  cau3lem  11675  climshftlemg  11863  subcn2  11872  serf0  11913  sqrt2irr  12735  pclemub  12861  prmpwdvds  12929  grpinveu  13622  dfgrp3mlem  13682  issubg4m  13781  tgcn  14934  tgcnp  14935  lmconst  14942  cnntr  14951  lmss  14972  txdis  15003  txlm  15005  blbas  15159  metss  15220  metcnp3  15237  iswomni0  16658