Theorem r19.21be 3136

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 3135	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓)
3	2	expcom 414	. 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))
4	3	rgen 3076	1 ⊢ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓)

Syntax hints:   → wi 4   ∈ wcel 2110  ∀wral 3066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-12 2175

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1787  df-ral 3071

This theorem is referenced by:  bnj580  32887