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Theorem bnj1019 32176
 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 1954 . 2 (∃𝑝((𝜃𝜒𝜂) ∧ 𝜏) ↔ ((𝜃𝜒𝜂) ∧ ∃𝑝𝜏))
2 bnj258 32103 . . 3 ((𝜃𝜒𝜏𝜂) ↔ ((𝜃𝜒𝜂) ∧ 𝜏))
32exbii 1849 . 2 (∃𝑝(𝜃𝜒𝜏𝜂) ↔ ∃𝑝((𝜃𝜒𝜂) ∧ 𝜏))
4 df-bnj17 32082 . 2 ((𝜃𝜒𝜂 ∧ ∃𝑝𝜏) ↔ ((𝜃𝜒𝜂) ∧ ∃𝑝𝜏))
51, 3, 43bitr4i 306 1 (∃𝑝(𝜃𝜒𝜏𝜂) ↔ (𝜃𝜒𝜂 ∧ ∃𝑝𝜏))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   ∧ w3a 1084  ∃wex 1781   ∧ w-bnj17 32081 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911 This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086  df-ex 1782  df-bnj17 32082 This theorem is referenced by:  bnj1018g  32360  bnj1018  32361  bnj1020  32362  bnj1021  32363
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