Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1019 Structured version   Visualization version   GIF version

Theorem bnj1019 29898
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 1905 . 2 (∃𝑝((𝜃𝜒𝜂) ∧ 𝜏) ↔ ((𝜃𝜒𝜂) ∧ ∃𝑝𝜏))
2 bnj258 29821 . . 3 ((𝜃𝜒𝜏𝜂) ↔ ((𝜃𝜒𝜂) ∧ 𝜏))
32exbii 1764 . 2 (∃𝑝(𝜃𝜒𝜏𝜂) ↔ ∃𝑝((𝜃𝜒𝜂) ∧ 𝜏))
4 df-bnj17 29800 . 2 ((𝜃𝜒𝜂 ∧ ∃𝑝𝜏) ↔ ((𝜃𝜒𝜂) ∧ ∃𝑝𝜏))
51, 3, 43bitr4i 291 1 (∃𝑝(𝜃𝜒𝜏𝜂) ↔ (𝜃𝜒𝜂 ∧ ∃𝑝𝜏))
Colors of variables: wff setvar class
Syntax hints:  wb 195  wa 383  w3a 1031  wex 1695  w-bnj17 29799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875
This theorem depends on definitions:  df-bi 196  df-an 385  df-3an 1033  df-ex 1696  df-bnj17 29800
This theorem is referenced by:  bnj1018  30080  bnj1020  30081  bnj1021  30082
  Copyright terms: Public domain W3C validator