Theorem pm11.6 41899

Description: Theorem *11.6 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 25-May-2011.)

Assertion

Ref	Expression
pm11.6	⊢ (∃𝑥(∃𝑦(𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ∃𝑦(∃𝑥(𝜑 ∧ 𝜒) ∧ 𝜓))

Distinct variable groups:   𝜓,𝑥   𝜒,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)   𝜒(𝑥)

Proof of Theorem pm11.6

Step	Hyp	Ref	Expression
1		excom 2164	. . 3 ⊢ (∃𝑥∃𝑦((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ∃𝑦∃𝑥((𝜑 ∧ 𝜓) ∧ 𝜒))
2		an32 642	. . . 4 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ((𝜑 ∧ 𝜒) ∧ 𝜓))
3	2	2exbii 1852	. . 3 ⊢ (∃𝑦∃𝑥((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ∃𝑦∃𝑥((𝜑 ∧ 𝜒) ∧ 𝜓))
4	1, 3	bitri 274	. 2 ⊢ (∃𝑥∃𝑦((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ∃𝑦∃𝑥((𝜑 ∧ 𝜒) ∧ 𝜓))
5		19.41v 1954	. . 3 ⊢ (∃𝑦((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ (∃𝑦(𝜑 ∧ 𝜓) ∧ 𝜒))
6	5	exbii 1851	. 2 ⊢ (∃𝑥∃𝑦((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ∃𝑥(∃𝑦(𝜑 ∧ 𝜓) ∧ 𝜒))
7		19.41v 1954	. . 3 ⊢ (∃𝑥((𝜑 ∧ 𝜒) ∧ 𝜓) ↔ (∃𝑥(𝜑 ∧ 𝜒) ∧ 𝜓))
8	7	exbii 1851	. 2 ⊢ (∃𝑦∃𝑥((𝜑 ∧ 𝜒) ∧ 𝜓) ↔ ∃𝑦(∃𝑥(𝜑 ∧ 𝜒) ∧ 𝜓))
9	4, 6, 8	3bitr3i 300	1 ⊢ (∃𝑥(∃𝑦(𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ∃𝑦(∃𝑥(𝜑 ∧ 𝜒) ∧ 𝜓))

Syntax hints:   ↔ wb 205   ∧ wa 395  ∃wex 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-11 2156

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784

This theorem is referenced by: (None)