Theorem ralimiaa 3086

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 413	. 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))
3	2	ralimia 3085	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2106  ∀wral 3064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812

This theorem depends on definitions:  df-bi 206  df-an 397  df-ral 3069

This theorem is referenced by:  ralrnmptw  6970  ralrnmpt  6972  tz7.48-2  8273  mptelixpg  8723  boxriin  8728  acnlem  9804  iundom2g  10296  konigthlem  10324  hashge2el2dif  14194  rlim2  15205  prdsbas3  17192  prdsdsval2  17195  ptbasfi  22732  ptunimpt  22746  voliun  24718  lgamgulmlem6  26183  riesz4i  30425  dmdbr6ati  30785  ctbssinf  35577