Theorem dfich2 47445

Description: Alternate definition of the property 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 47433	. 2 ⊢ ([𝑥⇄𝑦]𝜑 ↔ ∀𝑥∀𝑦([𝑥 / 𝑧][𝑦 / 𝑥][𝑧 / 𝑦]𝜑 ↔ 𝜑))
2		nfs1v 2156	. . . . . . 7 ⊢ Ⅎ𝑦[𝑏 / 𝑦]𝜑
3	2	nfsbv 2330	. . . . . 6 ⊢ Ⅎ𝑦[𝑎 / 𝑥][𝑏 / 𝑦]𝜑
4	3	nfsbv 2330	. . . . 5 ⊢ Ⅎ𝑦[𝑥 / 𝑏][𝑎 / 𝑥][𝑏 / 𝑦]𝜑
5		nfv 1914	. . . . 5 ⊢ Ⅎ𝑎𝜑
6	4, 5	sbbib 2364	. . . 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 1819	. . 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 2102	. . . . 5 ⊢ ([𝑦 / 𝑎][𝑥 / 𝑏][𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑥 / 𝑧][𝑦 / 𝑥][𝑧 / 𝑦]𝜑)
9	8	bibi1i 338	. . . 4 ⊢ (([𝑦 / 𝑎][𝑥 / 𝑏][𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ 𝜑) ↔ ([𝑥 / 𝑧][𝑦 / 𝑥][𝑧 / 𝑦]𝜑 ↔ 𝜑))
10	9	2albii 1820	. . 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 2159	. . . 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 2156	. . . . . 6 ⊢ Ⅎ𝑥[𝑎 / 𝑥][𝑏 / 𝑦]𝜑
13		nfv 1914	. . . . . 6 ⊢ Ⅎ𝑏[𝑎 / 𝑦]𝜑
14	12, 13	sbbib 2364	. . . . 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 1819	. . . 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 275	. . 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 301	. 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 275	1 ⊢ ([𝑥⇄𝑦]𝜑 ↔ ∀𝑎∀𝑏([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑥][𝑎 / 𝑦]𝜑))

Syntax hints:   ↔ wb 206  ∀wal 1538  [wsb 2064  [wich 47432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-10 2141  ax-11 2157  ax-12 2177

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-ich 47433

This theorem is referenced by:  ichcom  47446  ichbi12i  47447  ichnfim  47451  ichnreuop  47459  ichreuopeq  47460