Theorem ralimiaa 3101

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 418	. 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))
3	2	ralimia 3099	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜓)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2146  ∀wral 3079

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841

This proof depends on definitions:  df-bi 210  df-an 402  df-ral 3080

This theorem is used by:  ralrnmptw  7102  ralrnmpt  7104  tz7.48-2  8463  mptelixpg  8974  boxriin  8979  acnlem  10137  iundom2g  10634  konigthlem  10663  hashge2el2dif  14635  rlim2  15675  prdsbas3  17664  prdsdsval2  17667  ptbasfi  23912  ptunimpt  23926  voliun  25887  lgamgulmlem6  27373  riesz4i  32677  dmdbr6ati  33037  ctbssinf  38469