Theorem r19.21be 2716

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 2713	. . 3 ⊢ ((φ ∧ x ∈ A) → ψ)
3	2	expcom 424	. 2 ⊢ (x ∈ A → (φ → ψ))
4	3	rgen 2680	1 ⊢ ∀x ∈ A (φ → ψ)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 1710  ∀wral 2615

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This proof depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-ral 2620

This theorem is used by: (None)