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Metamath Proof Explorer

Theorem 2reu5lem2

Description: Lemma for 2reu5 . (Contributed by Alexander van der Vekens, 17-Jun-2017)

Ref Expression
Assertion 2reu5lem2 ( ∀ 𝑥 ∈ 𝐴 ∃* 𝑦 ∈ 𝐵 𝜑 ↔ ∀ 𝑥 ∃* 𝑦 ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑 ) )

Proof

Step Hyp Ref Expression
1 df-rmo ⊢ ( ∃* 𝑦 ∈ 𝐵 𝜑 ↔ ∃* 𝑦 ( 𝑦 ∈ 𝐵 ∧ 𝜑 ) )
2 1 ralbii ⊢ ( ∀ 𝑥 ∈ 𝐴 ∃* 𝑦 ∈ 𝐵 𝜑 ↔ ∀ 𝑥 ∈ 𝐴 ∃* 𝑦 ( 𝑦 ∈ 𝐵 ∧ 𝜑 ) )
3 df-ral ⊢ ( ∀ 𝑥 ∈ 𝐴 ∃* 𝑦 ( 𝑦 ∈ 𝐵 ∧ 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∃* 𝑦 ( 𝑦 ∈ 𝐵 ∧ 𝜑 ) ) )
4 moanimv ⊢ ( ∃* 𝑦 ( 𝑥 ∈ 𝐴 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝜑 ) ) ↔ ( 𝑥 ∈ 𝐴 → ∃* 𝑦 ( 𝑦 ∈ 𝐵 ∧ 𝜑 ) ) )
5 4 bicomi ⊢ ( ( 𝑥 ∈ 𝐴 → ∃* 𝑦 ( 𝑦 ∈ 𝐵 ∧ 𝜑 ) ) ↔ ∃* 𝑦 ( 𝑥 ∈ 𝐴 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝜑 ) ) )
6 3anass ⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑 ) ↔ ( 𝑥 ∈ 𝐴 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝜑 ) ) )
7 6 bicomi ⊢ ( ( 𝑥 ∈ 𝐴 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝜑 ) ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑 ) )
8 7 mobii ⊢ ( ∃* 𝑦 ( 𝑥 ∈ 𝐴 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝜑 ) ) ↔ ∃* 𝑦 ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑 ) )
9 5 8 bitri ⊢ ( ( 𝑥 ∈ 𝐴 → ∃* 𝑦 ( 𝑦 ∈ 𝐵 ∧ 𝜑 ) ) ↔ ∃* 𝑦 ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑 ) )
10 9 albii ⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∃* 𝑦 ( 𝑦 ∈ 𝐵 ∧ 𝜑 ) ) ↔ ∀ 𝑥 ∃* 𝑦 ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑 ) )
11 3 10 bitri ⊢ ( ∀ 𝑥 ∈ 𝐴 ∃* 𝑦 ( 𝑦 ∈ 𝐵 ∧ 𝜑 ) ↔ ∀ 𝑥 ∃* 𝑦 ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑 ) )
12 2 11 bitri ⊢ ( ∀ 𝑥 ∈ 𝐴 ∃* 𝑦 ∈ 𝐵 𝜑 ↔ ∀ 𝑥 ∃* 𝑦 ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑 ) )