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Theorem bnj1019 31950
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Assertion
Ref Expression
bnj1019 (∃𝑝(𝜃𝜒𝜏𝜂) ↔ (𝜃𝜒𝜂 ∧ ∃𝑝𝜏))
Distinct variable groups:   𝜒,𝑝   𝜂,𝑝   𝜃,𝑝
Allowed substitution hint:   𝜏(𝑝)

Proof of Theorem bnj1019
StepHypRef Expression
1 19.42v 1945 . 2 (∃𝑝((𝜃𝜒𝜂) ∧ 𝜏) ↔ ((𝜃𝜒𝜂) ∧ ∃𝑝𝜏))
2 bnj258 31877 . . 3 ((𝜃𝜒𝜏𝜂) ↔ ((𝜃𝜒𝜂) ∧ 𝜏))
32exbii 1839 . 2 (∃𝑝(𝜃𝜒𝜏𝜂) ↔ ∃𝑝((𝜃𝜒𝜂) ∧ 𝜏))
4 df-bnj17 31856 . 2 ((𝜃𝜒𝜂 ∧ ∃𝑝𝜏) ↔ ((𝜃𝜒𝜂) ∧ ∃𝑝𝜏))
51, 3, 43bitr4i 304 1 (∃𝑝(𝜃𝜒𝜏𝜂) ↔ (𝜃𝜒𝜂 ∧ ∃𝑝𝜏))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  w3a 1079  wex 1771  w-bnj17 31855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902
This theorem depends on definitions:  df-bi 208  df-an 397  df-3an 1081  df-ex 1772  df-bnj17 31856
This theorem is referenced by:  bnj1018  32133  bnj1020  32134  bnj1021  32135
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