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Metamath Proof Explorer

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 ) )