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Theorem bnj1521 29978
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 29922 . 2 (𝜒 → ∃𝑥(𝑥𝐵𝜑))
3 bnj1521.2 . 2 (𝜃 ↔ (𝜒𝑥𝐵𝜑))
4 bnj1521.3 . 2 (𝜒 → ∀𝑥𝜒)
52, 3, 4bnj1345 29952 1 (𝜒 → ∃𝑥𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  w3a 1030  wal 1472  wex 1694  wcel 1976  wrex 2893
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 2032
This theorem depends on definitions:  df-bi 195  df-an 384  df-3an 1032  df-ex 1695  df-nf 1700  df-rex 2898
This theorem is referenced by:  bnj1204  30137  bnj1311  30149  bnj1398  30159  bnj1408  30161  bnj1450  30175  bnj1312  30183  bnj1501  30192  bnj1523  30196
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