Theorem rgen2 2711

Description: Generalization rule for restricted quantification. (Contributed by NM, 30-May-1999.)

Hypothesis

Ref	Expression
rgen2.1	⊢ ((x ∈ A ∧ y ∈ B) → φ)

Assertion

Ref	Expression
rgen2	⊢ ∀x ∈ A ∀y ∈ B φ

Distinct variable groups:   x,y   y,A

Allowed substitution hints:   φ(x,y)   A(x)   B(x,y)

Proof of Theorem rgen2

Step	Hyp	Ref	Expression
1		rgen2.1	. . 3 ⊢ ((x ∈ A ∧ y ∈ B) → φ)
2	1	ralrimiva 2698	. 2 ⊢ (x ∈ A → ∀y ∈ B φ)
3	2	rgen 2680	1 ⊢ ∀x ∈ A ∀y ∈ B φ

Syntax hints:   → wi 4   ∧ wa 358   ∈ 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-nf 1545  df-ral 2620

This theorem is referenced by:  rgen3  2712