Theorem r19.21be 1269

Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.)

Hypothesis

Ref	Expression
r19.21be.1	⊢ (φ → ∀x ∈ A ψ)

Assertion

Ref	Expression
r19.21be	⊢ ∀x ∈ A (φ → ψ)

Proof of Theorem r19.21be

Step	Hyp	Ref	Expression
1		r19.21be.1	. . . . 5 ⊢ (φ → ∀x ∈ A ψ)
2	1	r19.21bi 1266	. . . 4 ⊢ ((φ ∧ x ∈ A) → ψ)
3	2	ancoms 334	. . 3 ⊢ ((x ∈ A ∧ φ) → ψ)
4	3	exp 291	. 2 ⊢ (x ∈ A → (φ → ψ))
5	4	rgen 1247	1 ⊢ ∀x ∈ A (φ → ψ)

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

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

This theorem depends on definitions:  df-bi 128  df-an 198  df-ral 1205

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