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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
StepHypRef 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))
42, 3ax-mp 5 . . 3 ((∃𝑥𝜑 → ∃*𝑥𝜑) → ∃*𝑥𝜑)
51, 4impbii 201 . 2 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑))
6 anclb 541 . 2 ((∃𝑥𝜑 → ∃*𝑥𝜑) ↔ (∃𝑥𝜑 → (∃𝑥𝜑 ∧ ∃*𝑥𝜑)))
7 df-eu 2587 . . . 4 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
87bicomi 216 . . 3 ((∃𝑥𝜑 ∧ ∃*𝑥𝜑) ↔ ∃!𝑥𝜑)
98imbi2i 328 . 2 ((∃𝑥𝜑 → (∃𝑥𝜑 ∧ ∃*𝑥𝜑)) ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
105, 6, 93bitri 289 1 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
 Colors of variables: wff setvar class 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)
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