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Theorem bnj1019 32472
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 1962 . 2 (∃𝑝((𝜃𝜒𝜂) ∧ 𝜏) ↔ ((𝜃𝜒𝜂) ∧ ∃𝑝𝜏))
2 bnj258 32399 . . 3 ((𝜃𝜒𝜏𝜂) ↔ ((𝜃𝜒𝜂) ∧ 𝜏))
32exbii 1855 . 2 (∃𝑝(𝜃𝜒𝜏𝜂) ↔ ∃𝑝((𝜃𝜒𝜂) ∧ 𝜏))
4 df-bnj17 32378 . 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 1089  wex 1787  w-bnj17 32377
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918
This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1091  df-ex 1788  df-bnj17 32378
This theorem is referenced by:  bnj1018g  32656  bnj1018  32657  bnj1020  32658  bnj1021  32659
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