Theorem r19.21be 2568

Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 21-Nov-1994.)

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	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)
2	1	r19.21bi 2565	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓)
3	2	expcom 116	. 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))
4	3	rgen 2530	1 ⊢ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓)

Syntax hints:   → wi 4   ∈ wcel 2148  ∀wral 2455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-gen 1449  ax-4 1510

This theorem depends on definitions:  df-bi 117  df-ral 2460

This theorem is referenced by: (None)