Theorem r19.20i2 1252

Description: Inference quantifying both antecedent and consequent.

Hypothesis

Ref	Expression
r19.20i2.1	⊢ ((x ∈ A → φ) → (x ∈ B → ψ))

Assertion

Ref	Expression
r19.20i2	⊢ (∀x ∈ A φ → ∀x ∈ B ψ)

Proof of Theorem r19.20i2

Step	Hyp	Ref	Expression
1		r19.20i2.1	. . 3 ⊢ ((x ∈ A → φ) → (x ∈ B → ψ))
2	1	19.20i 691	. 2 ⊢ (∀x(x ∈ A → φ) → ∀x(x ∈ B → ψ))
3		df-ral 1205	. 2 ⊢ (∀x ∈ A φ ↔ ∀x(x ∈ A → φ))
4		df-ral 1205	. 2 ⊢ (∀x ∈ B ψ ↔ ∀x(x ∈ B → ψ))
5	2, 3, 4	3imtr4 192	1 ⊢ (∀x ∈ A φ → ∀x ∈ B ψ)

Syntax hints:   → wi 2  ∀wal 672   ∈ wcel 1092  ∀wral 1201

This theorem is referenced by:  r19.20i 1253  ralcom3 1315  omex 3475  kmlem1 3580

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677

This theorem depends on definitions:  df-bi 128  df-ral 1205

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