Theorem r19.21bi 2713

Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 20-Nov-1994.)

Hypothesis

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

Assertion

Ref	Expression
r19.21bi	⊢ ((φ ∧ x ∈ A) → ψ)

Proof of Theorem r19.21bi

Step	Hyp	Ref	Expression
1		r19.21bi.1	. . . 4 ⊢ (φ → ∀x ∈ A ψ)
2		df-ral 2620	. . . 4 ⊢ (∀x ∈ A ψ ↔ ∀x(x ∈ A → ψ))
3	1, 2	sylib 188	. . 3 ⊢ (φ → ∀x(x ∈ A → ψ))
4	3	19.21bi 1758	. 2 ⊢ (φ → (x ∈ A → ψ))
5	4	imp 418	1 ⊢ ((φ ∧ x ∈ A) → ψ)

Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540   ∈ wcel 1710  ∀wral 2615

This theorem was proved from 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 theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-ral 2620

This theorem is referenced by:  rspec2  2714  rspec3  2715  r19.21be  2716  phidisjnn  4616