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Theorem bnj1340 29954
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1340.1 (𝜓 → ∃𝑥𝜃)
bnj1340.2 (𝜒 ↔ (𝜓𝜃))
bnj1340.3 (𝜓 → ∀𝑥𝜓)
Assertion
Ref Expression
bnj1340 (𝜓 → ∃𝑥𝜒)

Proof of Theorem bnj1340
StepHypRef Expression
1 bnj1340.3 . . 3 (𝜓 → ∀𝑥𝜓)
2 bnj1340.1 . . 3 (𝜓 → ∃𝑥𝜃)
31, 2bnj596 29876 . 2 (𝜓 → ∃𝑥(𝜓𝜃))
4 bnj1340.2 . 2 (𝜒 ↔ (𝜓𝜃))
53, 4bnj1198 29926 1 (𝜓 → ∃𝑥𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  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-ex 1695  df-nf 1700
This theorem is referenced by:  bnj1450  30178
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