Theorem ralimiaa 2494

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 114	. 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))
3	2	ralimia 2493	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ∧ wa 103   ∈ wcel 1480  ∀wral 2416

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425

This theorem depends on definitions:  df-bi 116  df-ral 2421

This theorem is referenced by:  ralrnmpt  5562  rexrnmpt  5563  acexmidlem2  5771  mptelixpg  6628