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Theorem bnj1521 34844
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 34787 . 2 (𝜒 → ∃𝑥(𝑥𝐵𝜑))
3 bnj1521.2 . 2 (𝜃 ↔ (𝜒𝑥𝐵𝜑))
4 bnj1521.3 . 2 (𝜒 → ∀𝑥𝜒)
52, 3, 4bnj1345 34817 1 (𝜒 → ∃𝑥𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  w3a 1086  wal 1535  wex 1776  wcel 2106  wrex 3068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-10 2139  ax-12 2175
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-ex 1777  df-nf 1781  df-rex 3069
This theorem is referenced by:  bnj1204  35005  bnj1311  35017  bnj1398  35027  bnj1408  35029  bnj1450  35043  bnj1312  35051  bnj1501  35060  bnj1523  35064
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