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Theorem moanimlem 2706
 Description: Factor out the common proof skeleton of moanimv 2707 and moanim 2708. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 24-Dec-2018.) Factor out common proof lines. (Revised by Wolf Lammen, 8-Feb-2023.)
Hypotheses
Ref Expression
moanimlem.1 (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥(𝜑𝜓)))
moanimlem.2 (∃𝑥(𝜑𝜓) → 𝜑)
Assertion
Ref Expression
moanimlem (∃*𝑥(𝜑𝜓) ↔ (𝜑 → ∃*𝑥𝜓))

Proof of Theorem moanimlem
StepHypRef Expression
1 moanimlem.1 . . 3 (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥(𝜑𝜓)))
21biimprcd 242 . 2 (∃*𝑥(𝜑𝜓) → (𝜑 → ∃*𝑥𝜓))
3 moanimlem.2 . . . . 5 (∃𝑥(𝜑𝜓) → 𝜑)
4 nexmo 2606 . . . . 5 (¬ ∃𝑥(𝜑𝜓) → ∃*𝑥(𝜑𝜓))
53, 4nsyl4 158 . . . 4 (¬ ∃*𝑥(𝜑𝜓) → 𝜑)
65con1i 147 . . 3 𝜑 → ∃*𝑥(𝜑𝜓))
7 moan 2621 . . 3 (∃*𝑥𝜓 → ∃*𝑥(𝜑𝜓))
86, 7ja 175 . 2 ((𝜑 → ∃*𝑥𝜓) → ∃*𝑥(𝜑𝜓))
92, 8impbii 201 1 (∃*𝑥(𝜑𝜓) ↔ (𝜑 → ∃*𝑥𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386  ∃wex 1878  ∃*wmo 2603 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075 This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1879  df-mo 2605 This theorem is referenced by:  moanimv  2707  moanim  2708
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