Theorem ralimiaa 3070

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 3068	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2113  ∀wral 3049

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 3050

This theorem is referenced by:  ralrnmptw  7037  ralrnmpt  7039  tz7.48-2  8371  mptelixpg  8871  boxriin  8876  acnlem  9956  iundom2g  10448  konigthlem  10477  hashge2el2dif  14401  rlim2  15417  prdsbas3  17399  prdsdsval2  17402  ptbasfi  23523  ptunimpt  23537  voliun  25509  lgamgulmlem6  26998  riesz4i  32087  dmdbr6ati  32447  ctbssinf  37550