Theorem exmoeuOLD 2651

Description: Obsolete proof of exmoeu 2621 as of 7-Oct-2022. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Wolf Lammen, 5-Dec-2018.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem exmoeuOLD

Step	Hyp	Ref	Expression
1		moeu 2623	. . . 4 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
2	1	biimpi 208	. . 3 ⊢ (∃*𝑥𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑))
3	2	com12 32	. 2 ⊢ (∃𝑥𝜑 → (∃*𝑥𝜑 → ∃!𝑥𝜑))
4		exmo 2602	. . . . 5 ⊢ (∃𝑥𝜑 ∨ ∃*𝑥𝜑)
5	4	ori 888	. . . 4 ⊢ (¬ ∃𝑥𝜑 → ∃*𝑥𝜑)
6	5	con1i 147	. . 3 ⊢ (¬ ∃*𝑥𝜑 → ∃𝑥𝜑)
7		euex 2617	. . 3 ⊢ (∃!𝑥𝜑 → ∃𝑥𝜑)
8	6, 7	ja 175	. 2 ⊢ ((∃*𝑥𝜑 → ∃!𝑥𝜑) → ∃𝑥𝜑)
9	3, 8	impbii 201	1 ⊢ (∃𝑥𝜑 ↔ (∃*𝑥𝜑 → ∃!𝑥𝜑))

Syntax hints:   → wi 4   ↔ wb 198  ∃wex 1875  ∃*wmo 2589  ∃!weu 2608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-ex 1876  df-mo 2591  df-eu 2609

This theorem is referenced by: (None)