Theorem rgen2w 2683

Description: Generalization rule for restricted quantification. Note that x and y needn't be distinct. (Contributed by NM, 18-Jun-2014.)

Hypothesis

Ref	Expression
rgenw.1	⊢ φ

Assertion

Ref	Expression
rgen2w	⊢ ∀x ∈ A ∀y ∈ B φ

Proof of Theorem rgen2w

Step	Hyp	Ref	Expression
1		rgenw.1	. . 3 ⊢ φ
2	1	rgenw 2682	. 2 ⊢ ∀y ∈ B φ
3	2	rgenw 2682	1 ⊢ ∀x ∈ A ∀y ∈ B φ

Syntax hints:  ∀wral 2615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546

This theorem depends on definitions:  df-bi 177  df-ral 2620

This theorem is referenced by:  fnmpt2i  5734