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Theorem bnj1275 29944
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1275.1 (𝜑 → ∃𝑥(𝜓𝜒))
bnj1275.2 (𝜑 → ∀𝑥𝜑)
Assertion
Ref Expression
bnj1275 (𝜑 → ∃𝑥(𝜑𝜓𝜒))

Proof of Theorem bnj1275
StepHypRef Expression
1 bnj1275.2 . . 3 (𝜑 → ∀𝑥𝜑)
2 bnj1275.1 . . 3 (𝜑 → ∃𝑥(𝜓𝜒))
31, 2bnj596 29876 . 2 (𝜑 → ∃𝑥(𝜑 ∧ (𝜓𝜒)))
4 3anass 1034 . 2 ((𝜑𝜓𝜒) ↔ (𝜑 ∧ (𝜓𝜒)))
53, 4bnj1198 29926 1 (𝜑 → ∃𝑥(𝜑𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030  wal 1472  wex 1694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-12 2033
This theorem depends on definitions:  df-bi 195  df-an 384  df-3an 1032  df-ex 1695  df-nf 1700
This theorem is referenced by:  bnj1345  29955  bnj1279  30146
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