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Theorem exan 1786
 Description: Place a conjunct in the scope of an existential quantifier. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 13-Jan-2018.) Reduce axiom dependencies. (Revised by BJ, 7-Jul-2021.) (Proof shortened by Wolf Lammen, 8-Oct-2021.)
Hypothesis
Ref Expression
exan.1 (∃𝑥𝜑𝜓)
Assertion
Ref Expression
exan 𝑥(𝜑𝜓)

Proof of Theorem exan
StepHypRef Expression
1 exan.1 . . 3 (∃𝑥𝜑𝜓)
21simpli 474 . 2 𝑥𝜑
31simpri 478 . . . 4 𝜓
4 pm3.21 464 . . . 4 (𝜓 → (𝜑 → (𝜑𝜓)))
53, 4ax-mp 5 . . 3 (𝜑 → (𝜑𝜓))
65eximi 1760 . 2 (∃𝑥𝜑 → ∃𝑥(𝜑𝜓))
72, 6ax-mp 5 1 𝑥(𝜑𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384  ∃wex 1702 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735 This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1703 This theorem is referenced by:  bm1.3ii  4775  ac6s6f  33952  fnchoice  39008
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