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Theorem bnj1397 32385
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 2290 . 2 (∃𝑥𝜓𝜓)
41, 3sylib 221 1 (𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1540  wex 1786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-10 2145  ax-12 2179
This theorem depends on definitions:  df-bi 210  df-ex 1787  df-nf 1791
This theorem is referenced by:  bnj1398  32585  bnj1408  32587  bnj1450  32601  bnj1501  32618
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