Theorem rgenw 2682

Description: Generalization rule for restricted quantification. (Contributed by NM, 18-Jun-2014.)

Hypothesis

Ref	Expression
rgenw.1	⊢ φ

Assertion

Ref	Expression
rgenw	⊢ ∀x ∈ A φ

Proof of Theorem rgenw

Step	Hyp	Ref	Expression
1		rgenw.1	. . 3 ⊢ φ
2	1	a1i 10	. 2 ⊢ (x ∈ A → φ)
3	2	rgen 2680	1 ⊢ ∀x ∈ A φ

Syntax hints:   ∈ 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

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

This theorem is referenced by:  rgen2w  2683  reuun1  3538  riinrab  4042  evenodddisj  4517  vfinspnn  4542  mpt2eq12  5663  fnmpti  5691  clos10  5888  fnpm  6009  frecxp  6315