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Metamath Proof Explorer

Theorem rgen2

Description: Generalization rule for restricted quantification, with two quantifiers. This theorem should be used in place of rgen2a since it depends on a smaller set of axioms. (Contributed by NM, 30-May-1999)

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

Proof

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