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Theorem bnj1340 30148
 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 30070 . 2 (𝜓 → ∃𝑥(𝜓𝜃))
4 bnj1340.2 . 2 (𝜒 ↔ (𝜓𝜃))
53, 4bnj1198 30120 1 (𝜓 → ∃𝑥𝜒)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473  ∃wex 1695 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-nf 1701 This theorem is referenced by:  bnj1450  30372
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