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Theorem pm11.6 38095
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 2039 . . 3 (∃𝑥𝑦((𝜑𝜓) ∧ 𝜒) ↔ ∃𝑦𝑥((𝜑𝜓) ∧ 𝜒))
2 an32 838 . . . 4 (((𝜑𝜓) ∧ 𝜒) ↔ ((𝜑𝜒) ∧ 𝜓))
322exbii 1772 . . 3 (∃𝑦𝑥((𝜑𝜓) ∧ 𝜒) ↔ ∃𝑦𝑥((𝜑𝜒) ∧ 𝜓))
41, 3bitri 264 . 2 (∃𝑥𝑦((𝜑𝜓) ∧ 𝜒) ↔ ∃𝑦𝑥((𝜑𝜒) ∧ 𝜓))
5 19.41v 1911 . . 3 (∃𝑦((𝜑𝜓) ∧ 𝜒) ↔ (∃𝑦(𝜑𝜓) ∧ 𝜒))
65exbii 1771 . 2 (∃𝑥𝑦((𝜑𝜓) ∧ 𝜒) ↔ ∃𝑥(∃𝑦(𝜑𝜓) ∧ 𝜒))
7 19.41v 1911 . . 3 (∃𝑥((𝜑𝜒) ∧ 𝜓) ↔ (∃𝑥(𝜑𝜒) ∧ 𝜓))
87exbii 1771 . 2 (∃𝑦𝑥((𝜑𝜒) ∧ 𝜓) ↔ ∃𝑦(∃𝑥(𝜑𝜒) ∧ 𝜓))
94, 6, 83bitr3i 290 1 (∃𝑥(∃𝑦(𝜑𝜓) ∧ 𝜒) ↔ ∃𝑦(∃𝑥(𝜑𝜒) ∧ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  wex 1701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-11 2031
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702
This theorem is referenced by: (None)
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