Theorem r19.43

Description: Restricted quantifier version of 19.43 . (Contributed by NM, 27-May-1998) (Proof shortened by Andrew Salmon, 30-May-2011)

	Ref	Expression
Assertion	r19.43	⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∨ 𝜓 ) ↔ ( ∃ 𝑥 ∈ 𝐴 𝜑 ∨ ∃ 𝑥 ∈ 𝐴 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		r19.35	⊢ ( ∃ 𝑥 ∈ 𝐴 ( ¬ 𝜑 → 𝜓 ) ↔ ( ∀ 𝑥 ∈ 𝐴 ¬ 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 ) )
2		df-or	⊢ ( ( 𝜑 ∨ 𝜓 ) ↔ ( ¬ 𝜑 → 𝜓 ) )
3	2	rexbii	⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∨ 𝜓 ) ↔ ∃ 𝑥 ∈ 𝐴 ( ¬ 𝜑 → 𝜓 ) )
4		df-or	⊢ ( ( ∃ 𝑥 ∈ 𝐴 𝜑 ∨ ∃ 𝑥 ∈ 𝐴 𝜓 ) ↔ ( ¬ ∃ 𝑥 ∈ 𝐴 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 ) )
5		ralnex	⊢ ( ∀ 𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∃ 𝑥 ∈ 𝐴 𝜑 )
6	5	imbi1i	⊢ ( ( ∀ 𝑥 ∈ 𝐴 ¬ 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 ) ↔ ( ¬ ∃ 𝑥 ∈ 𝐴 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 ) )
7	4 6	bitr4i	⊢ ( ( ∃ 𝑥 ∈ 𝐴 𝜑 ∨ ∃ 𝑥 ∈ 𝐴 𝜓 ) ↔ ( ∀ 𝑥 ∈ 𝐴 ¬ 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 ) )
8	1 3 7	3bitr4i	⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∨ 𝜓 ) ↔ ( ∃ 𝑥 ∈ 𝐴 𝜑 ∨ ∃ 𝑥 ∈ 𝐴 𝜓 ) )