Theorem dfich2ai 43688

Description: Obsolete version of dfich2 43687 as of 18-Sep-2023. The definition df-ich 43680 of the propery of a wff 𝜑 that the setvar variables 𝑥 and 𝑦 are interchangeable implies the alternate definition dfich2 43687. (Contributed by AV, 6-Aug-2023.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
dfich2ai	⊢ (∀𝑥∀𝑦([𝑥 / 𝑧][𝑦 / 𝑥][𝑧 / 𝑦]𝜑 ↔ 𝜑) → ([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑥][𝑎 / 𝑦]𝜑))

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

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem dfich2ai

Step	Hyp	Ref	Expression
1		sbco4 2283	. . . . 5 ⊢ ([𝑦 / 𝑎][𝑥 / 𝑏][𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑥 / 𝑧][𝑦 / 𝑥][𝑧 / 𝑦]𝜑)
2	1	bicomi 226	. . . 4 ⊢ ([𝑥 / 𝑧][𝑦 / 𝑥][𝑧 / 𝑦]𝜑 ↔ [𝑦 / 𝑎][𝑥 / 𝑏][𝑎 / 𝑥][𝑏 / 𝑦]𝜑)
3	2	bibi1i 341	. . 3 ⊢ (([𝑥 / 𝑧][𝑦 / 𝑥][𝑧 / 𝑦]𝜑 ↔ 𝜑) ↔ ([𝑦 / 𝑎][𝑥 / 𝑏][𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ 𝜑))
4	3	2albii 1820	. 2 ⊢ (∀𝑥∀𝑦([𝑥 / 𝑧][𝑦 / 𝑥][𝑧 / 𝑦]𝜑 ↔ 𝜑) ↔ ∀𝑥∀𝑦([𝑦 / 𝑎][𝑥 / 𝑏][𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ 𝜑))
5		nfs1v 2159	. . . . . . 7 ⊢ Ⅎ𝑦[𝑏 / 𝑦]𝜑
6	5	nfsbv 2348	. . . . . 6 ⊢ Ⅎ𝑦[𝑎 / 𝑥][𝑏 / 𝑦]𝜑
7	6	nfsbv 2348	. . . . 5 ⊢ Ⅎ𝑦[𝑥 / 𝑏][𝑎 / 𝑥][𝑏 / 𝑦]𝜑
8		nfv 1914	. . . . 5 ⊢ Ⅎ𝑎𝜑
9	7, 8	sbbib 2379	. . . 4 ⊢ (∀𝑦([𝑦 / 𝑎][𝑥 / 𝑏][𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ 𝜑) ↔ ∀𝑎([𝑥 / 𝑏][𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑎 / 𝑦]𝜑))
10	9	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))
11		alcom 2162	. . . . 5 ⊢ (∀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 2159	. . . . . . 7 ⊢ Ⅎ𝑥[𝑎 / 𝑥][𝑏 / 𝑦]𝜑
13		nfv 1914	. . . . . . 7 ⊢ Ⅎ𝑏[𝑎 / 𝑦]𝜑
14	12, 13	sbbib 2379	. . . . . 6 ⊢ (∀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	. . . . 5 ⊢ (∀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	. . . 4 ⊢ (∀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		2sp 2184	. . . 4 ⊢ (∀𝑎∀𝑏([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑥][𝑎 / 𝑦]𝜑) → ([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑥][𝑎 / 𝑦]𝜑))
18	16, 17	sylbi 219	. . 3 ⊢ (∀𝑥∀𝑎([𝑥 / 𝑏][𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑎 / 𝑦]𝜑) → ([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑥][𝑎 / 𝑦]𝜑))
19	10, 18	sylbi 219	. 2 ⊢ (∀𝑥∀𝑦([𝑦 / 𝑎][𝑥 / 𝑏][𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ 𝜑) → ([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑥][𝑎 / 𝑦]𝜑))
20	4, 19	sylbi 219	1 ⊢ (∀𝑥∀𝑦([𝑥 / 𝑧][𝑦 / 𝑥][𝑧 / 𝑦]𝜑 ↔ 𝜑) → ([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑥][𝑎 / 𝑦]𝜑))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1534  [wsb 2068

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 1969  ax-7 2014  ax-10 2144  ax-11 2160  ax-12 2176

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069

This theorem is referenced by:  dfich2OLD  43690