Theorem rgen2w 2553

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 2552	. 2 ⊢ ∀𝑦 ∈ 𝐵 𝜑
3	2	rgenw 2552	1 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑

Syntax hints:  ∀wral 2475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-gen 1463

This theorem depends on definitions:  df-bi 117  df-ral 2480

This theorem is referenced by:  fnmpoi  6261  ixxf  9973  fzf  10087  rexfiuz  11154  eltx  14495  txcnp  14507  txcnmpt  14509  txrest  14512  txlm  14515