Theorem ralimiaa 2935

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 449	. 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))
3	2	ralimia 2934	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977  ∀wral 2896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728

This theorem depends on definitions:  df-bi 196  df-an 385  df-ral 2901

This theorem is referenced by:  ralrnmpt  6276  tz7.48-2  7424  mptelixpg  7831  boxriin  7836  acnlem  8754  iundom2g  9241  konigthlem  9269  hashge2el2dif  13117  rlim2  14075  prdsbas3  15964  prdsdsval2  15967  ptbasfi  21194  ptunimpt  21208  voliun  23129  lgamgulmlem6  24560  riesz4i  28306  dmdbr6ati  28666