Theorem ralimiaa 3065

Description: Inference quantifying both antecedent and consequent. (Contributed by NM, 4-Aug-2007.)

Hypothesis

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

Assertion

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

Proof of Theorem ralimiaa

Step	Hyp	Ref	Expression
1		ralimiaa.1	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓)
2	1	ex 412	. 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))
3	2	ralimia 3063	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109  ∀wral 3044

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809

This theorem depends on definitions:  df-bi 207  df-an 396  df-ral 3045

This theorem is referenced by:  ralrnmptw  7048  ralrnmpt  7050  tz7.48-2  8387  mptelixpg  8885  boxriin  8890  acnlem  9977  iundom2g  10469  konigthlem  10497  hashge2el2dif  14421  rlim2  15438  prdsbas3  17420  prdsdsval2  17423  ptbasfi  23444  ptunimpt  23458  voliun  25431  lgamgulmlem6  26920  riesz4i  31965  dmdbr6ati  32325  ctbssinf  37367