Theorem 19.43OLD

Description: Obsolete proof of 19.43 . Do not delete as it is referenced on the mmrecent.html page and in conventions-labels . (Contributed by NM, 5-Aug-1993) (Proof modification is discouraged.) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		ioran	⊢ ( ¬ ( 𝜑 ∨ 𝜓 ) ↔ ( ¬ 𝜑 ∧ ¬ 𝜓 ) )
2	1	albii	⊢ ( ∀ 𝑥 ¬ ( 𝜑 ∨ 𝜓 ) ↔ ∀ 𝑥 ( ¬ 𝜑 ∧ ¬ 𝜓 ) )
3		19.26	⊢ ( ∀ 𝑥 ( ¬ 𝜑 ∧ ¬ 𝜓 ) ↔ ( ∀ 𝑥 ¬ 𝜑 ∧ ∀ 𝑥 ¬ 𝜓 ) )
4		alnex	⊢ ( ∀ 𝑥 ¬ 𝜑 ↔ ¬ ∃ 𝑥 𝜑 )
5		alnex	⊢ ( ∀ 𝑥 ¬ 𝜓 ↔ ¬ ∃ 𝑥 𝜓 )
6	4 5	anbi12i	⊢ ( ( ∀ 𝑥 ¬ 𝜑 ∧ ∀ 𝑥 ¬ 𝜓 ) ↔ ( ¬ ∃ 𝑥 𝜑 ∧ ¬ ∃ 𝑥 𝜓 ) )
7	2 3 6	3bitri	⊢ ( ∀ 𝑥 ¬ ( 𝜑 ∨ 𝜓 ) ↔ ( ¬ ∃ 𝑥 𝜑 ∧ ¬ ∃ 𝑥 𝜓 ) )
8	7	notbii	⊢ ( ¬ ∀ 𝑥 ¬ ( 𝜑 ∨ 𝜓 ) ↔ ¬ ( ¬ ∃ 𝑥 𝜑 ∧ ¬ ∃ 𝑥 𝜓 ) )
9		df-ex	⊢ ( ∃ 𝑥 ( 𝜑 ∨ 𝜓 ) ↔ ¬ ∀ 𝑥 ¬ ( 𝜑 ∨ 𝜓 ) )
10		oran	⊢ ( ( ∃ 𝑥 𝜑 ∨ ∃ 𝑥 𝜓 ) ↔ ¬ ( ¬ ∃ 𝑥 𝜑 ∧ ¬ ∃ 𝑥 𝜓 ) )
11	8 9 10	3bitr4i	⊢ ( ∃ 𝑥 ( 𝜑 ∨ 𝜓 ) ↔ ( ∃ 𝑥 𝜑 ∨ ∃ 𝑥 𝜓 ) )