Theorem raaan

Description: Rearrange restricted quantifiers. (Contributed by NM, 26-Oct-2010)

	Ref	Expression
Hypotheses	raaan.1	⊢ Ⅎ 𝑦 𝜑
	raaan.2	⊢ Ⅎ 𝑥 𝜓
Assertion	raaan	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) ↔ ( ∀ 𝑥 ∈ 𝐴 𝜑 ∧ ∀ 𝑦 ∈ 𝐴 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		raaan.1	⊢ Ⅎ 𝑦 𝜑
2		raaan.2	⊢ Ⅎ 𝑥 𝜓
3		rzal	⊢ ( 𝐴 = ∅ → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) )
4		rzal	⊢ ( 𝐴 = ∅ → ∀ 𝑥 ∈ 𝐴 𝜑 )
5		rzal	⊢ ( 𝐴 = ∅ → ∀ 𝑦 ∈ 𝐴 𝜓 )
6		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 ) ) )
7	3 4 5 6	syl12anc	⊢ ( 𝐴 = ∅ → ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) ↔ ( ∀ 𝑥 ∈ 𝐴 𝜑 ∧ ∀ 𝑦 ∈ 𝐴 𝜓 ) ) )
8	1	r19.28z	⊢ ( 𝐴 ≠ ∅ → ( ∀ 𝑦 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜑 ∧ ∀ 𝑦 ∈ 𝐴 𝜓 ) ) )
9	8	ralbidv	⊢ ( 𝐴 ≠ ∅ → ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) ↔ ∀ 𝑥 ∈ 𝐴 ( 𝜑 ∧ ∀ 𝑦 ∈ 𝐴 𝜓 ) ) )
10		nfcv	⊢ Ⅎ 𝑥 𝐴
11	10 2	nfralw	⊢ Ⅎ 𝑥 ∀ 𝑦 ∈ 𝐴 𝜓
12	11	r19.27z	⊢ ( 𝐴 ≠ ∅ → ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 ∧ ∀ 𝑦 ∈ 𝐴 𝜓 ) ↔ ( ∀ 𝑥 ∈ 𝐴 𝜑 ∧ ∀ 𝑦 ∈ 𝐴 𝜓 ) ) )
13	9 12	bitrd	⊢ ( 𝐴 ≠ ∅ → ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) ↔ ( ∀ 𝑥 ∈ 𝐴 𝜑 ∧ ∀ 𝑦 ∈ 𝐴 𝜓 ) ) )
14	7 13	pm2.61ine	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) ↔ ( ∀ 𝑥 ∈ 𝐴 𝜑 ∧ ∀ 𝑦 ∈ 𝐴 𝜓 ) )