Theorem moeuOLD 2618

Description: Obsolete proof of moeu 2603 as of 14-Oct-2022. (Contributed by NM, 8-Mar-1995.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
moeuOLD	⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))

Proof of Theorem moeuOLD

Step	Hyp	Ref	Expression
1		ax-1 6	. . 3 ⊢ (∃*𝑥𝜑 → (∃𝑥𝜑 → ∃*𝑥𝜑))
2		nexmo 2552	. . . 4 ⊢ (¬ ∃𝑥𝜑 → ∃*𝑥𝜑)
3		pm2.6 183	. . . 4 ⊢ ((¬ ∃𝑥𝜑 → ∃*𝑥𝜑) → ((∃𝑥𝜑 → ∃*𝑥𝜑) → ∃*𝑥𝜑))
4	2, 3	ax-mp 5	. . 3 ⊢ ((∃𝑥𝜑 → ∃*𝑥𝜑) → ∃*𝑥𝜑)
5	1, 4	impbii 201	. 2 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑))
6		anclb 541	. 2 ⊢ ((∃𝑥𝜑 → ∃*𝑥𝜑) ↔ (∃𝑥𝜑 → (∃𝑥𝜑 ∧ ∃*𝑥𝜑)))
7		df-eu 2587	. . . 4 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
8	7	bicomi 216	. . 3 ⊢ ((∃𝑥𝜑 ∧ ∃*𝑥𝜑) ↔ ∃!𝑥𝜑)
9	8	imbi2i 328	. 2 ⊢ ((∃𝑥𝜑 → (∃𝑥𝜑 ∧ ∃*𝑥𝜑)) ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
10	5, 6, 9	3bitri 289	1 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386  ∃wex 1823  ∃*wmo 2549  ∃!weu 2586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021

This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1824  df-mo 2551  df-eu 2587

This theorem is referenced by: (None)