Theorem nexmoOLD 2549

Description: Obsolete version of nexmo 2548 as of 16-Oct-2022. (Contributed by BJ, 30-Sep-2022.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
nexmoOLD	⊢ (¬ ∃𝑥𝜑 → ∃*𝑥𝜑)

Proof of Theorem nexmoOLD

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		alnex 1744	. 2 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
2		pm2.21 121	. . . . . . 7 ⊢ (¬ 𝜑 → (𝜑 → 𝑥 = 𝑦))
3	2	alrimiv 1886	. . . . . 6 ⊢ (¬ 𝜑 → ∀𝑦(𝜑 → 𝑥 = 𝑦))
4	3	alimi 1774	. . . . 5 ⊢ (∀𝑥 ¬ 𝜑 → ∀𝑥∀𝑦(𝜑 → 𝑥 = 𝑦))
5		alcom 2095	. . . . 5 ⊢ (∀𝑥∀𝑦(𝜑 → 𝑥 = 𝑦) ↔ ∀𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
6	4, 5	sylib 210	. . . 4 ⊢ (∀𝑥 ¬ 𝜑 → ∀𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
7	6	19.2d 1935	. . 3 ⊢ (∀𝑥 ¬ 𝜑 → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
8		df-mo 2547	. . 3 ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
9	7, 8	sylibr 226	. 2 ⊢ (∀𝑥 ¬ 𝜑 → ∃*𝑥𝜑)
10	1, 9	sylbir 227	1 ⊢ (¬ ∃𝑥𝜑 → ∃*𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1505  ∃wex 1742  ∃*wmo 2545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-11 2093

This theorem depends on definitions:  df-bi 199  df-ex 1743  df-mo 2547

This theorem is referenced by: (None)