Theorem ichf 47925

Description: Setvar variables are interchangeable in a wff they are not free in. (Contributed by SN, 23-Nov-2023.)

Hypotheses

Ref	Expression
ichf.1	⊢ Ⅎ𝑥𝜑
ichf.2	⊢ Ⅎ𝑦𝜑

Assertion

Ref	Expression
ichf	⊢ [𝑥⇄𝑦]𝜑

Proof of Theorem ichf

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ichf.2	. . . . . . . 8 ⊢ Ⅎ𝑦𝜑
2	1	sbf 2282	. . . . . . 7 ⊢ ([𝑎 / 𝑦]𝜑 ↔ 𝜑)
3	2	sbbii 2087	. . . . . 6 ⊢ ([𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)
4		ichf.1	. . . . . . 7 ⊢ Ⅎ𝑥𝜑
5	4	sbf 2282	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)
6	3, 5	bitri 276	. . . . 5 ⊢ ([𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ 𝜑)
7	6	sbbii 2087	. . . 4 ⊢ ([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ [𝑥 / 𝑎]𝜑)
8		sbv 2099	. . . 4 ⊢ ([𝑥 / 𝑎]𝜑 ↔ 𝜑)
9	7, 8	bitri 276	. . 3 ⊢ ([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ 𝜑)
10	9	gen2 1803	. 2 ⊢ ∀𝑥∀𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ 𝜑)
11		df-ich 47921	. 2 ⊢ ([𝑥⇄𝑦]𝜑 ↔ ∀𝑥∀𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ 𝜑))
12	10, 11	mpbir 232	1 ⊢ [𝑥⇄𝑦]𝜑

Syntax hints:   ↔ wb 207  ∀wal 1545  Ⅎwnf 1790  [wsb 2073  [wich 47920

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-12 2189

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-nf 1791  df-sb 2074  df-ich 47921

This theorem is referenced by:  ich2al  47942  ich2ex  47943