Theorem ralrimdva 2545

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 114	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒)))
3	2	com23 78	. 2 ⊢ (𝜑 → (𝜓 → (𝑥 ∈ 𝐴 → 𝜒)))
4	3	ralrimdv 2544	1 ⊢ (𝜑 → (𝜓 → ∀𝑥 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4   ∧ wa 103   ∈ wcel 2136  ∀wral 2443

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-4 1498  ax-17 1514

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-ral 2448

This theorem is referenced by:  ralxfrd  4439  isoselem  5787  isosolem  5791  findcard  6850  nnsub  8892  supinfneg  9529  infsupneg  9530  ublbneg  9547  expnlbnd2  10576  cau3lem  11052  climshftlemg  11239  subcn2  11248  serf0  11289  sqrt2irr  12090  pclemub  12215  prmpwdvds  12281  tgcn  12808  tgcnp  12809  lmconst  12816  cnntr  12825  lmss  12846  txdis  12877  txlm  12879  blbas  13033  metss  13094  metcnp3  13111  iswomni0  13890