Theorem icheq 43669

Description: In an equality of setvar variables, the setvar variables are interchangeable. (Contributed by AV, 29-Jul-2023.)

Assertion

Ref	Expression
icheq	⊢ [𝑥⇄𝑦]𝑥 = 𝑦

Distinct variable group:   𝑥,𝑦

Proof of Theorem icheq

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equsb3r 2110	. . . . 5 ⊢ ([𝑧 / 𝑦]𝑥 = 𝑦 ↔ 𝑥 = 𝑧)
2	1	2sbbii 2082	. . . 4 ⊢ ([𝑥 / 𝑧][𝑦 / 𝑥][𝑧 / 𝑦]𝑥 = 𝑦 ↔ [𝑥 / 𝑧][𝑦 / 𝑥]𝑥 = 𝑧)
3		equsb3 2109	. . . . 5 ⊢ ([𝑦 / 𝑥]𝑥 = 𝑧 ↔ 𝑦 = 𝑧)
4	3	sbbii 2081	. . . 4 ⊢ ([𝑥 / 𝑧][𝑦 / 𝑥]𝑥 = 𝑧 ↔ [𝑥 / 𝑧]𝑦 = 𝑧)
5		equsb3r 2110	. . . . 5 ⊢ ([𝑥 / 𝑧]𝑦 = 𝑧 ↔ 𝑦 = 𝑥)
6		equcom 2025	. . . . 5 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
7	5, 6	bitri 277	. . . 4 ⊢ ([𝑥 / 𝑧]𝑦 = 𝑧 ↔ 𝑥 = 𝑦)
8	2, 4, 7	3bitri 299	. . 3 ⊢ ([𝑥 / 𝑧][𝑦 / 𝑥][𝑧 / 𝑦]𝑥 = 𝑦 ↔ 𝑥 = 𝑦)
9	8	gen2 1797	. 2 ⊢ ∀𝑥∀𝑦([𝑥 / 𝑧][𝑦 / 𝑥][𝑧 / 𝑦]𝑥 = 𝑦 ↔ 𝑥 = 𝑦)
10		df-ich 43655	. 2 ⊢ ([𝑥⇄𝑦]𝑥 = 𝑦 ↔ ∀𝑥∀𝑦([𝑥 / 𝑧][𝑦 / 𝑥][𝑧 / 𝑦]𝑥 = 𝑦 ↔ 𝑥 = 𝑦))
11	9, 10	mpbir 233	1 ⊢ [𝑥⇄𝑦]𝑥 = 𝑦

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

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-sb 2070  df-ich 43655

This theorem is referenced by: (None)