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Theorem bnj1275 32258
 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 32190 . 2 (𝜑 → ∃𝑥(𝜑 ∧ (𝜓𝜒)))
4 3anass 1092 . 2 ((𝜑𝜓𝜒) ↔ (𝜑 ∧ (𝜓𝜒)))
53, 4bnj1198 32240 1 (𝜑 → ∃𝑥(𝜑𝜓𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084  ∀wal 1536  ∃wex 1781 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-12 2175 This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086  df-ex 1782  df-nf 1786 This theorem is referenced by:  bnj1345  32269  bnj1279  32463
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