Theorem rspec2 2713

Description: Specialization rule for restricted quantification. (Contributed by NM, 20-Nov-1994.)

Hypothesis

Ref	Expression
rspec2.1	⊢ ∀x ∈ A ∀y ∈ B φ

Assertion

Ref	Expression
rspec2	⊢ ((x ∈ A ∧ y ∈ B) → φ)

Proof of Theorem rspec2

Step	Hyp	Ref	Expression
1		rspec2.1	. . 3 ⊢ ∀x ∈ A ∀y ∈ B φ
2	1	rspec 2678	. 2 ⊢ (x ∈ A → ∀y ∈ B φ)
3	2	r19.21bi 2712	1 ⊢ ((x ∈ A ∧ y ∈ B) → φ)

Syntax hints:   → wi 4   ∧ wa 358   ∈ wcel 1710  ∀wral 2614

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 2619

This theorem is referenced by:  rspec3  2714