Theorem dfich2 43662

Description: Alternate definition of the propery of a wff 𝜑 that the setvar variables 𝑥 and 𝑦 are interchangeable. (Contributed by AV and WL, 6-Aug-2023.)

Assertion

Ref	Expression
dfich2	⊢ ([𝑥⇄𝑦]𝜑 ↔ ∀𝑎∀𝑏([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑥][𝑎 / 𝑦]𝜑))

Distinct variable groups:   𝑎,𝑏,𝜑   𝑥,𝑎,𝑦,𝑏

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem dfich2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ich 43655	. 2 ⊢ ([𝑥⇄𝑦]𝜑 ↔ ∀𝑥∀𝑦([𝑥 / 𝑧][𝑦 / 𝑥][𝑧 / 𝑦]𝜑 ↔ 𝜑))
2		nfs1v 2160	. . . . . . 7 ⊢ Ⅎ𝑦[𝑏 / 𝑦]𝜑
3	2	nfsbv 2349	. . . . . 6 ⊢ Ⅎ𝑦[𝑎 / 𝑥][𝑏 / 𝑦]𝜑
4	3	nfsbv 2349	. . . . 5 ⊢ Ⅎ𝑦[𝑥 / 𝑏][𝑎 / 𝑥][𝑏 / 𝑦]𝜑
5		nfv 1915	. . . . 5 ⊢ Ⅎ𝑎𝜑
6	4, 5	sbbib 2380	. . . 4 ⊢ (∀𝑦([𝑦 / 𝑎][𝑥 / 𝑏][𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ 𝜑) ↔ ∀𝑎([𝑥 / 𝑏][𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑎 / 𝑦]𝜑))
7	6	albii 1820	. . 3 ⊢ (∀𝑥∀𝑦([𝑦 / 𝑎][𝑥 / 𝑏][𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ 𝜑) ↔ ∀𝑥∀𝑎([𝑥 / 𝑏][𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑎 / 𝑦]𝜑))
8		sbco4 2284	. . . . 5 ⊢ ([𝑦 / 𝑎][𝑥 / 𝑏][𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑥 / 𝑧][𝑦 / 𝑥][𝑧 / 𝑦]𝜑)
9	8	bibi1i 341	. . . 4 ⊢ (([𝑦 / 𝑎][𝑥 / 𝑏][𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ 𝜑) ↔ ([𝑥 / 𝑧][𝑦 / 𝑥][𝑧 / 𝑦]𝜑 ↔ 𝜑))
10	9	2albii 1821	. . 3 ⊢ (∀𝑥∀𝑦([𝑦 / 𝑎][𝑥 / 𝑏][𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ 𝜑) ↔ ∀𝑥∀𝑦([𝑥 / 𝑧][𝑦 / 𝑥][𝑧 / 𝑦]𝜑 ↔ 𝜑))
11		alcom 2163	. . . 4 ⊢ (∀𝑥∀𝑎([𝑥 / 𝑏][𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑎 / 𝑦]𝜑) ↔ ∀𝑎∀𝑥([𝑥 / 𝑏][𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑎 / 𝑦]𝜑))
12		nfs1v 2160	. . . . . 6 ⊢ Ⅎ𝑥[𝑎 / 𝑥][𝑏 / 𝑦]𝜑
13		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑏[𝑎 / 𝑦]𝜑
14	12, 13	sbbib 2380	. . . . 5 ⊢ (∀𝑥([𝑥 / 𝑏][𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑎 / 𝑦]𝜑) ↔ ∀𝑏([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑥][𝑎 / 𝑦]𝜑))
15	14	albii 1820	. . . 4 ⊢ (∀𝑎∀𝑥([𝑥 / 𝑏][𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑎 / 𝑦]𝜑) ↔ ∀𝑎∀𝑏([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑥][𝑎 / 𝑦]𝜑))
16	11, 15	bitri 277	. . 3 ⊢ (∀𝑥∀𝑎([𝑥 / 𝑏][𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑎 / 𝑦]𝜑) ↔ ∀𝑎∀𝑏([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑥][𝑎 / 𝑦]𝜑))
17	7, 10, 16	3bitr3i 303	. 2 ⊢ (∀𝑥∀𝑦([𝑥 / 𝑧][𝑦 / 𝑥][𝑧 / 𝑦]𝜑 ↔ 𝜑) ↔ ∀𝑎∀𝑏([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑥][𝑎 / 𝑦]𝜑))
18	1, 17	bitri 277	1 ⊢ ([𝑥⇄𝑦]𝜑 ↔ ∀𝑎∀𝑏([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑥][𝑎 / 𝑦]𝜑))

Syntax hints:   ↔ wb 208  ∀wal 1535  [wsb 2069  [wich 43654

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-11 2161  ax-12 2177

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-ich 43655

This theorem is referenced by:  ichcom  43666  ichbi12i  43667  ichnfim  43673  ichan  43679  ichnreuop  43683  ichreuopeq  43684