Theorem ralimia 2687

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 2686	1 ⊢ (∀x ∈ A φ → ∀x ∈ A ψ)

Syntax hints:   → wi 4   ∈ wcel 1710  ∀wral 2614

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557

This theorem depends on definitions:  df-bi 177  df-ral 2619

This theorem is referenced by:  ralimiaa  2688  ralimi  2689  r19.12  2727  rr19.3v  2980  rr19.28v  2981  ffvresb  5431