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Metamath Proof Explorer

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 ) )