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Metamath Proof Explorer

Theorem reeanlem

Description: Lemma factoring out common proof steps of reeanv and reean . (Contributed by Wolf Lammen, 20-Aug-2023)

Ref Expression
Hypothesis reeanlem.1 ⊢ ( ∃ 𝑥 ∃ 𝑦 ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝜓 ) ) ↔ ( ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ∧ ∃ 𝑦 ( 𝑦 ∈ 𝐵 ∧ 𝜓 ) ) )
Assertion reeanlem ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ( 𝜑 ∧ 𝜓 ) ↔ ( ∃ 𝑥 ∈ 𝐴 𝜑 ∧ ∃ 𝑦 ∈ 𝐵 𝜓 ) )

Proof

Step Hyp Ref Expression
1 reeanlem.1 ⊢ ( ∃ 𝑥 ∃ 𝑦 ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝜓 ) ) ↔ ( ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ∧ ∃ 𝑦 ( 𝑦 ∈ 𝐵 ∧ 𝜓 ) ) )
2 an4 ⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝜑 ∧ 𝜓 ) ) ↔ ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝜓 ) ) )
3 2 2exbii ⊢ ( ∃ 𝑥 ∃ 𝑦 ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝜑 ∧ 𝜓 ) ) ↔ ∃ 𝑥 ∃ 𝑦 ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝜓 ) ) )
4 3 1 bitri ⊢ ( ∃ 𝑥 ∃ 𝑦 ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝜑 ∧ 𝜓 ) ) ↔ ( ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ∧ ∃ 𝑦 ( 𝑦 ∈ 𝐵 ∧ 𝜓 ) ) )
5 r2ex ⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ( 𝜑 ∧ 𝜓 ) ↔ ∃ 𝑥 ∃ 𝑦 ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝜑 ∧ 𝜓 ) ) )
6 df-rex ⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) )
7 df-rex ⊢ ( ∃ 𝑦 ∈ 𝐵 𝜓 ↔ ∃ 𝑦 ( 𝑦 ∈ 𝐵 ∧ 𝜓 ) )
8 6 7 anbi12i ⊢ ( ( ∃ 𝑥 ∈ 𝐴 𝜑 ∧ ∃ 𝑦 ∈ 𝐵 𝜓 ) ↔ ( ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ∧ ∃ 𝑦 ( 𝑦 ∈ 𝐵 ∧ 𝜓 ) ) )
9 4 5 8 3bitr4i ⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ( 𝜑 ∧ 𝜓 ) ↔ ( ∃ 𝑥 ∈ 𝐴 𝜑 ∧ ∃ 𝑦 ∈ 𝐵 𝜓 ) )