Theorem r19.20i 1253

Description: Inference quantifying both antecedent and consequent.

Hypothesis

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

Assertion

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

Proof of Theorem r19.20i

Step	Hyp	Ref	Expression
1		r19.20i.1	. . 3 ⊢ (x ∈ A → (φ → ψ))
2	1	a2i 8	. 2 ⊢ ((x ∈ A → φ) → (x ∈ A → ψ))
3	2	r19.20i2 1252	1 ⊢ (∀x ∈ A φ → ∀x ∈ A ψ)

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

This theorem is referenced by:  r19.20si 1254  r19.12 1281  tz7.48-2 2995  tz9.12lem3 3505  aceq6a 3564  kmlem11 3590  arch 4521  climunii 4883  hlimunii 5143  spanun 5450

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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