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 ) )