Theorem ralimiaa 3072

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 3070	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2113  ∀wral 3051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810

This theorem depends on definitions:  df-bi 207  df-an 396  df-ral 3052

This theorem is referenced by:  ralrnmptw  7039  ralrnmpt  7041  tz7.48-2  8373  mptelixpg  8873  boxriin  8878  acnlem  9958  iundom2g  10450  konigthlem  10479  hashge2el2dif  14403  rlim2  15419  prdsbas3  17401  prdsdsval2  17404  ptbasfi  23525  ptunimpt  23539  voliun  25511  lgamgulmlem6  27000  riesz4i  32138  dmdbr6ati  32498  ctbssinf  37611