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Theorem exan 1948
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 472 . 2 𝑥𝜑
31simpri 475 . . . 4 𝜓
4 pm3.21 459 . . . 4 (𝜓 → (𝜑 → (𝜑𝜓)))
53, 4ax-mp 5 . . 3 (𝜑 → (𝜑𝜓))
65eximi 1919 . 2 (∃𝑥𝜑 → ∃𝑥(𝜑𝜓))
72, 6ax-mp 5 1 𝑥(𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wex 1859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894
This theorem depends on definitions:  df-bi 198  df-an 385  df-ex 1860
This theorem is referenced by:  bm1.3ii  4991  ac6s6f  34310  fnchoice  39700
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