Theorem ralrimdva 2586

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 2585	1 ⊢ (𝜑 → (𝜓 → ∀𝑥 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2176  ∀wral 2484

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 1470  ax-gen 1472  ax-4 1533  ax-17 1549

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-ral 2489

This theorem is referenced by:  ralxfrd  4509  isoselem  5889  isosolem  5893  findcard  6985  nnsub  9075  supinfneg  9716  infsupneg  9717  ublbneg  9734  expnlbnd2  10810  cau3lem  11425  climshftlemg  11613  subcn2  11622  serf0  11663  sqrt2irr  12484  pclemub  12610  prmpwdvds  12678  grpinveu  13370  dfgrp3mlem  13430  issubg4m  13529  tgcn  14680  tgcnp  14681  lmconst  14688  cnntr  14697  lmss  14718  txdis  14749  txlm  14751  blbas  14905  metss  14966  metcnp3  14983  iswomni0  15994