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Metamath Proof Explorer

Theorem r2allem

Description: Lemma factoring out common proof steps of r2alf and r2al . Introduced to reduce dependencies on axioms. (Contributed by Wolf Lammen, 9-Jan-2020)

Ref Expression
Hypothesis r2allem.1 ⊢ ( ∀ 𝑦 ( 𝑥 ∈ 𝐴 → ( 𝑦 ∈ 𝐵 → 𝜑 ) ) ↔ ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ( 𝑦 ∈ 𝐵 → 𝜑 ) ) )
Assertion r2allem ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜑 ↔ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) → 𝜑 ) )

Proof

Step Hyp Ref Expression
1 r2allem.1 ⊢ ( ∀ 𝑦 ( 𝑥 ∈ 𝐴 → ( 𝑦 ∈ 𝐵 → 𝜑 ) ) ↔ ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ( 𝑦 ∈ 𝐵 → 𝜑 ) ) )
2 df-ral ⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜑 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ∈ 𝐵 𝜑 ) )
3 impexp ⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) → 𝜑 ) ↔ ( 𝑥 ∈ 𝐴 → ( 𝑦 ∈ 𝐵 → 𝜑 ) ) )
4 3 albii ⊢ ( ∀ 𝑦 ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) → 𝜑 ) ↔ ∀ 𝑦 ( 𝑥 ∈ 𝐴 → ( 𝑦 ∈ 𝐵 → 𝜑 ) ) )
5 df-ral ⊢ ( ∀ 𝑦 ∈ 𝐵 𝜑 ↔ ∀ 𝑦 ( 𝑦 ∈ 𝐵 → 𝜑 ) )
6 5 imbi2i ⊢ ( ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ∈ 𝐵 𝜑 ) ↔ ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ( 𝑦 ∈ 𝐵 → 𝜑 ) ) )
7 1 4 6 3bitr4i ⊢ ( ∀ 𝑦 ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) → 𝜑 ) ↔ ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ∈ 𝐵 𝜑 ) )
8 7 albii ⊢ ( ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) → 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ∈ 𝐵 𝜑 ) )
9 2 8 bitr4i ⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜑 ↔ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) → 𝜑 ) )