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Theorem bnj1397 30159
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1397.1 (𝜑 → ∃𝑥𝜓)
bnj1397.2 (𝜓 → ∀𝑥𝜓)
Assertion
Ref Expression
bnj1397 (𝜑𝜓)

Proof of Theorem bnj1397
StepHypRef Expression
1 bnj1397.1 . 2 (𝜑 → ∃𝑥𝜓)
2 bnj1397.2 . . 3 (𝜓 → ∀𝑥𝜓)
3219.9h 2106 . 2 (∃𝑥𝜓𝜓)
41, 3sylib 207 1 (𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1473  wex 1695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034
This theorem depends on definitions:  df-bi 196  df-ex 1696  df-nf 1701
This theorem is referenced by:  bnj1398  30356  bnj1408  30358  bnj1450  30372  bnj1501  30389
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