Theorem 19.43OLD 1884

Description: Obsolete proof of 19.43 1883. Do not delete as it is referenced on the mmrecent.html 1883 page and in conventions-labels 28182. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem 19.43OLD

Step	Hyp	Ref	Expression
1		ioran 980	. . . . 5 ⊢ (¬ (𝜑 ∨ 𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓))
2	1	albii 1820	. . . 4 ⊢ (∀𝑥 ¬ (𝜑 ∨ 𝜓) ↔ ∀𝑥(¬ 𝜑 ∧ ¬ 𝜓))
3		19.26 1871	. . . 4 ⊢ (∀𝑥(¬ 𝜑 ∧ ¬ 𝜓) ↔ (∀𝑥 ¬ 𝜑 ∧ ∀𝑥 ¬ 𝜓))
4		alnex 1782	. . . . 5 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
5		alnex 1782	. . . . 5 ⊢ (∀𝑥 ¬ 𝜓 ↔ ¬ ∃𝑥𝜓)
6	4, 5	anbi12i 628	. . . 4 ⊢ ((∀𝑥 ¬ 𝜑 ∧ ∀𝑥 ¬ 𝜓) ↔ (¬ ∃𝑥𝜑 ∧ ¬ ∃𝑥𝜓))
7	2, 3, 6	3bitri 299	. . 3 ⊢ (∀𝑥 ¬ (𝜑 ∨ 𝜓) ↔ (¬ ∃𝑥𝜑 ∧ ¬ ∃𝑥𝜓))
8	7	notbii 322	. 2 ⊢ (¬ ∀𝑥 ¬ (𝜑 ∨ 𝜓) ↔ ¬ (¬ ∃𝑥𝜑 ∧ ¬ ∃𝑥𝜓))
9		df-ex 1781	. 2 ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ ¬ ∀𝑥 ¬ (𝜑 ∨ 𝜓))
10		oran 986	. 2 ⊢ ((∃𝑥𝜑 ∨ ∃𝑥𝜓) ↔ ¬ (¬ ∃𝑥𝜑 ∧ ¬ ∃𝑥𝜓))
11	8, 9, 10	3bitr4i 305	1 ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))

Syntax hints:  ¬ wn 3   ↔ wb 208   ∧ wa 398   ∨ wo 843  ∀wal 1535  ∃wex 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781

This theorem is referenced by: (None)