Theorem rgen2a 1264

Description: Generalization rule for restricted quantification.

Hypothesis

Ref	Expression
rgen2a.1	⊢ ((x ∈ A ∧ y ∈ B) → φ)

Assertion

Ref	Expression
rgen2a	⊢ ∀x ∈ A ∀y ∈ B φ

Distinct variable group(s):   x,y   y,A

Proof of Theorem rgen2a

Step	Hyp	Ref	Expression
1		rgen2a.1	. . . 4 ⊢ ((x ∈ A ∧ y ∈ B) → φ)
2	1	exp 291	. . 3 ⊢ (x ∈ A → (y ∈ B → φ))
3	2	r19.21aiv 1259	. 2 ⊢ (x ∈ A → ∀y ∈ B φ)
4	3	rgen 1247	1 ⊢ ∀x ∈ A ∀y ∈ B φ

Syntax hints:   → wi 2   ∧ wa 196   ∈ wcel 1092  ∀wral 1201

This theorem is referenced by:  f1stres 3096  ruclem13 4897  helch 5151  hsn0elch 5155  shscl 5282  shintcl 5294

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677  ax-17 925

This theorem depends on definitions:  df-bi 128  df-an 198  df-ral 1205

metamath.org