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Theorem bnj1275 32506
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 32438 . 2 (𝜑 → ∃𝑥(𝜑 ∧ (𝜓𝜒)))
4 3anass 1097 . 2 ((𝜑𝜓𝜒) ↔ (𝜑 ∧ (𝜓𝜒)))
53, 4bnj1198 32488 1 (𝜑 → ∃𝑥(𝜑𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1089  wal 1541  wex 1787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-10 2141  ax-12 2175
This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1091  df-ex 1788  df-nf 1792
This theorem is referenced by:  bnj1345  32517  bnj1279  32711
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