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Theorem bnj1521 32415
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1521.1 (𝜒 → ∃𝑥𝐵 𝜑)
bnj1521.2 (𝜃 ↔ (𝜒𝑥𝐵𝜑))
bnj1521.3 (𝜒 → ∀𝑥𝜒)
Assertion
Ref Expression
bnj1521 (𝜒 → ∃𝑥𝜃)

Proof of Theorem bnj1521
StepHypRef Expression
1 bnj1521.1 . . 3 (𝜒 → ∃𝑥𝐵 𝜑)
21bnj1196 32358 . 2 (𝜒 → ∃𝑥(𝑥𝐵𝜑))
3 bnj1521.2 . 2 (𝜃 ↔ (𝜒𝑥𝐵𝜑))
4 bnj1521.3 . 2 (𝜒 → ∀𝑥𝜒)
52, 3, 4bnj1345 32388 1 (𝜒 → ∃𝑥𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  w3a 1088  wal 1540  wex 1786  wcel 2114  wrex 3055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-10 2145  ax-12 2179
This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1090  df-ex 1787  df-nf 1791  df-rex 3060
This theorem is referenced by:  bnj1204  32576  bnj1311  32588  bnj1398  32598  bnj1408  32600  bnj1450  32614  bnj1312  32622  bnj1501  32631  bnj1523  32635
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