Theorem ichf 48056

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 2305	. . . . . . 7 ⊢ ([𝑎 / 𝑦]𝜑 ↔ 𝜑)
3	2	sbbii 2109	. . . . . 6 ⊢ ([𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)
4		ichf.1	. . . . . . 7 ⊢ Ⅎ𝑥𝜑
5	4	sbf 2305	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)
6	3, 5	bitri 277	. . . . 5 ⊢ ([𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ 𝜑)
7	6	sbbii 2109	. . . 4 ⊢ ([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ [𝑥 / 𝑎]𝜑)
8		sbv 2121	. . . 4 ⊢ ([𝑥 / 𝑎]𝜑 ↔ 𝜑)
9	7, 8	bitri 277	. . 3 ⊢ ([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ 𝜑)
10	9	gen2 1816	. 2 ⊢ ∀𝑥∀𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ 𝜑)
11		df-ich 48052	. 2 ⊢ ([𝑥⇄𝑦]𝜑 ↔ ∀𝑥∀𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ 𝜑))
12	10, 11	mpbir 233	1 ⊢ [𝑥⇄𝑦]𝜑

Syntax hints:   ↔ wb 208  ∀wal 1558  Ⅎwnf 1803  [wsb 2090  [wich 48051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-12 2212

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1800  df-nf 1804  df-sb 2091  df-ich 48052

This theorem is referenced by:  ich2al  48073  ich2ex  48074