Theorem ralimia 2688

Description: Inference quantifying both antecedent and consequent. (Contributed by NM, 19-Jul-1996.)

Hypothesis

Ref	Expression
ralimia.1	⊢ (x ∈ A → (φ → ψ))

Assertion

Ref	Expression
ralimia	⊢ (∀x ∈ A φ → ∀x ∈ A ψ)

Proof of Theorem ralimia

Step	Hyp	Ref	Expression
1		ralimia.1	. . 3 ⊢ (x ∈ A → (φ → ψ))
2	1	a2i 12	. 2 ⊢ ((x ∈ A → φ) → (x ∈ A → ψ))
3	2	ralimi2 2687	1 ⊢ (∀x ∈ A φ → ∀x ∈ A ψ)

This proof depends on syntax axioms:   → wi 4   ∈ 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-ral 2620

This theorem is used by:  ralimiaa  2689  ralimi  2690  r19.12  2728  rr19.3v  2981  rr19.28v  2982  ffvresb  5432