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Theorem exan 1670
 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.)
Hypothesis
Ref Expression
exan.1 (∃𝑥𝜑𝜓)
Assertion
Ref Expression
exan 𝑥(𝜑𝜓)

Proof of Theorem exan
StepHypRef Expression
1 hbe1 1472 . . . 4 (∃𝑥𝜑 → ∀𝑥𝑥𝜑)
2119.28h 1539 . . 3 (∀𝑥(∃𝑥𝜑𝜓) ↔ (∃𝑥𝜑 ∧ ∀𝑥𝜓))
3 exan.1 . . 3 (∃𝑥𝜑𝜓)
42, 3mpgbi 1429 . 2 (∃𝑥𝜑 ∧ ∀𝑥𝜓)
5 19.29r 1598 . 2 ((∃𝑥𝜑 ∧ ∀𝑥𝜓) → ∃𝑥(𝜑𝜓))
64, 5ax-mp 5 1 𝑥(𝜑𝜓)
 Colors of variables: wff set class Syntax hints:   ∧ wa 103  ∀wal 1330  ∃wex 1469 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1487  ax-ial 1511 This theorem depends on definitions:  df-bi 116 This theorem is referenced by:  bm1.3ii  4081  bdbm1.3ii  13412
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