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Theorem moanimlem 2704
Description: Factor out the common proof skeleton of moanimv 2705 and moanim 2706. (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 253 . 2 (∃*𝑥(𝜑𝜓) → (𝜑 → ∃*𝑥𝜓))
3 moanimlem.2 . . . 4 (∃𝑥(𝜑𝜓) → 𝜑)
4 nexmo 2623 . . . 4 (¬ ∃𝑥(𝜑𝜓) → ∃*𝑥(𝜑𝜓))
53, 4nsyl5 162 . . 3 𝜑 → ∃*𝑥(𝜑𝜓))
6 moan 2635 . . 3 (∃*𝑥𝜓 → ∃*𝑥(𝜑𝜓))
75, 6ja 189 . 2 ((𝜑 → ∃*𝑥𝜓) → ∃*𝑥(𝜑𝜓))
82, 7impbii 212 1 (∃*𝑥(𝜑𝜓) ↔ (𝜑 → ∃*𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wex 1781  ∃*wmo 2620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-mo 2622
This theorem is referenced by:  moanimv  2705  moanim  2706
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