Theorem raaanv

Description: Rearrange restricted quantifiers. (Contributed by NM, 11-Mar-1997)

	Ref	Expression
Assertion	raaanv	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) ↔ ( ∀ 𝑥 ∈ 𝐴 𝜑 ∧ ∀ 𝑦 ∈ 𝐴 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		rzal	⊢ ( 𝐴 = ∅ → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) )
2		rzal	⊢ ( 𝐴 = ∅ → ∀ 𝑥 ∈ 𝐴 𝜑 )
3		rzal	⊢ ( 𝐴 = ∅ → ∀ 𝑦 ∈ 𝐴 𝜓 )
4		pm5.1	⊢ ( ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) ∧ ( ∀ 𝑥 ∈ 𝐴 𝜑 ∧ ∀ 𝑦 ∈ 𝐴 𝜓 ) ) → ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) ↔ ( ∀ 𝑥 ∈ 𝐴 𝜑 ∧ ∀ 𝑦 ∈ 𝐴 𝜓 ) ) )
5	1 2 3 4	syl12anc	⊢ ( 𝐴 = ∅ → ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) ↔ ( ∀ 𝑥 ∈ 𝐴 𝜑 ∧ ∀ 𝑦 ∈ 𝐴 𝜓 ) ) )
6		r19.28zv	⊢ ( 𝐴 ≠ ∅ → ( ∀ 𝑦 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜑 ∧ ∀ 𝑦 ∈ 𝐴 𝜓 ) ) )
7	6	ralbidv	⊢ ( 𝐴 ≠ ∅ → ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) ↔ ∀ 𝑥 ∈ 𝐴 ( 𝜑 ∧ ∀ 𝑦 ∈ 𝐴 𝜓 ) ) )
8		r19.27zv	⊢ ( 𝐴 ≠ ∅ → ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 ∧ ∀ 𝑦 ∈ 𝐴 𝜓 ) ↔ ( ∀ 𝑥 ∈ 𝐴 𝜑 ∧ ∀ 𝑦 ∈ 𝐴 𝜓 ) ) )
9	7 8	bitrd	⊢ ( 𝐴 ≠ ∅ → ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) ↔ ( ∀ 𝑥 ∈ 𝐴 𝜑 ∧ ∀ 𝑦 ∈ 𝐴 𝜓 ) ) )
10	5 9	pm2.61ine	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) ↔ ( ∀ 𝑥 ∈ 𝐴 𝜑 ∧ ∀ 𝑦 ∈ 𝐴 𝜓 ) )