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Metamath Proof Explorer

Theorem 19.43

Description: Theorem 19.43 of Margaris p. 90. (Contributed by NM, 12-Mar-1993) (Proof shortened by Wolf Lammen, 27-Jun-2014)

Ref Expression
Assertion 19.43 ( ∃ 𝑥 ( 𝜑 ∨ 𝜓 ) ↔ ( ∃ 𝑥 𝜑 ∨ ∃ 𝑥 𝜓 ) )

Proof

Step Hyp Ref Expression
1 df-or ⊢ ( ( 𝜑 ∨ 𝜓 ) ↔ ( ¬ 𝜑 → 𝜓 ) )
2 1 exbii ⊢ ( ∃ 𝑥 ( 𝜑 ∨ 𝜓 ) ↔ ∃ 𝑥 ( ¬ 𝜑 → 𝜓 ) )
3 19.35 ⊢ ( ∃ 𝑥 ( ¬ 𝜑 → 𝜓 ) ↔ ( ∀ 𝑥 ¬ 𝜑 → ∃ 𝑥 𝜓 ) )
4 alnex ⊢ ( ∀ 𝑥 ¬ 𝜑 ↔ ¬ ∃ 𝑥 𝜑 )
5 4 imbi1i ⊢ ( ( ∀ 𝑥 ¬ 𝜑 → ∃ 𝑥 𝜓 ) ↔ ( ¬ ∃ 𝑥 𝜑 → ∃ 𝑥 𝜓 ) )
6 2 3 5 3bitri ⊢ ( ∃ 𝑥 ( 𝜑 ∨ 𝜓 ) ↔ ( ¬ ∃ 𝑥 𝜑 → ∃ 𝑥 𝜓 ) )
7 df-or ⊢ ( ( ∃ 𝑥 𝜑 ∨ ∃ 𝑥 𝜓 ) ↔ ( ¬ ∃ 𝑥 𝜑 → ∃ 𝑥 𝜓 ) )
8 6 7 bitr4i ⊢ ( ∃ 𝑥 ( 𝜑 ∨ 𝜓 ) ↔ ( ∃ 𝑥 𝜑 ∨ ∃ 𝑥 𝜓 ) )