Theorem r19.21be 3259

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 3258	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓)
3	2	expcom 419	. 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))
4	3	rgen 3082	1 ⊢ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147  ∀wral 3080

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-ral 3081

This theorem is used by:  bnj580  35610