Theorem ralimi2 2375

Description: Inference quantifying both antecedent and consequent. (Contributed by NM, 22-Feb-2004.)

Hypothesis

Ref	Expression
ralimi2.1	⊢ ((x ∈ A → φ) → (x ∈ B → ψ))

Assertion

Ref	Expression
ralimi2	⊢ (∀x ∈ A φ → ∀x ∈ B ψ)

Proof of Theorem ralimi2

Step	Hyp	Ref	Expression
1		ralimi2.1	. . 3 ⊢ ((x ∈ A → φ) → (x ∈ B → ψ))
2	1	alimi 1341	. 2 ⊢ (∀x(x ∈ A → φ) → ∀x(x ∈ B → ψ))
3		df-ral 2305	. 2 ⊢ (∀x ∈ A φ ↔ ∀x(x ∈ A → φ))
4		df-ral 2305	. 2 ⊢ (∀x ∈ B ψ ↔ ∀x(x ∈ B → ψ))
5	2, 3, 4	3imtr4i 190	1 ⊢ (∀x ∈ A φ → ∀x ∈ B ψ)

Syntax hints:   → wi 4  ∀wal 1240   ∈ wcel 1390  ∀wral 2300

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335

This theorem depends on definitions:  df-bi 110  df-ral 2305

This theorem is referenced by:  ralimia  2376  ralcom3  2471  bj-nntrans  9411  bj-findis  9439