Theorem ralimiaa 2689

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 423	. 2 ⊢ (x ∈ A → (φ → ψ))
3	2	ralimia 2688	1 ⊢ (∀x ∈ A φ → ∀x ∈ A ψ)

This proof depends on syntax axioms:   → wi 4   ∧ wa 358   ∈ wcel 1710  ∀wral 2615

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557

This proof depends on definitions:  df-bi 177  df-an 360  df-ral 2620

This theorem is used by: (None)