Users' Mathboxes Mathbox for Andrew Salmon < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pm11.6 Structured version   Visualization version   GIF version

Theorem pm11.6 37614
Description: Theorem *11.6 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 25-May-2011.)
Assertion
Ref Expression
pm11.6 (∃𝑥(∃𝑦(𝜑𝜓) ∧ 𝜒) ↔ ∃𝑦(∃𝑥(𝜑𝜒) ∧ 𝜓))
Distinct variable groups:   𝜓,𝑥   𝜒,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)   𝜒(𝑥)

Proof of Theorem pm11.6
StepHypRef Expression
1 excom 2029 . . 3 (∃𝑥𝑦((𝜑𝜓) ∧ 𝜒) ↔ ∃𝑦𝑥((𝜑𝜓) ∧ 𝜒))
2 an32 835 . . . 4 (((𝜑𝜓) ∧ 𝜒) ↔ ((𝜑𝜒) ∧ 𝜓))
322exbii 1765 . . 3 (∃𝑦𝑥((𝜑𝜓) ∧ 𝜒) ↔ ∃𝑦𝑥((𝜑𝜒) ∧ 𝜓))
41, 3bitri 263 . 2 (∃𝑥𝑦((𝜑𝜓) ∧ 𝜒) ↔ ∃𝑦𝑥((𝜑𝜒) ∧ 𝜓))
5 19.41v 1901 . . 3 (∃𝑦((𝜑𝜓) ∧ 𝜒) ↔ (∃𝑦(𝜑𝜓) ∧ 𝜒))
65exbii 1764 . 2 (∃𝑥𝑦((𝜑𝜓) ∧ 𝜒) ↔ ∃𝑥(∃𝑦(𝜑𝜓) ∧ 𝜒))
7 19.41v 1901 . . 3 (∃𝑥((𝜑𝜒) ∧ 𝜓) ↔ (∃𝑥(𝜑𝜒) ∧ 𝜓))
87exbii 1764 . 2 (∃𝑦𝑥((𝜑𝜒) ∧ 𝜓) ↔ ∃𝑦(∃𝑥(𝜑𝜒) ∧ 𝜓))
94, 6, 83bitr3i 289 1 (∃𝑥(∃𝑦(𝜑𝜓) ∧ 𝜒) ↔ ∃𝑦(∃𝑥(𝜑𝜒) ∧ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 195  wa 383  wex 1695
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  ax-11 2021
This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator