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Theorem bnj1521 31253
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 31197 . 2 (𝜒 → ∃𝑥(𝑥𝐵𝜑))
3 bnj1521.2 . 2 (𝜃 ↔ (𝜒𝑥𝐵𝜑))
4 bnj1521.3 . 2 (𝜒 → ∀𝑥𝜒)
52, 3, 4bnj1345 31227 1 (𝜒 → ∃𝑥𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1070  wal 1628  wex 1851  wcel 2144  wrex 3061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-10 2173  ax-12 2202
This theorem depends on definitions:  df-bi 197  df-an 383  df-3an 1072  df-ex 1852  df-nf 1857  df-rex 3066
This theorem is referenced by:  bnj1204  31412  bnj1311  31424  bnj1398  31434  bnj1408  31436  bnj1450  31450  bnj1312  31458  bnj1501  31467  bnj1523  31471
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