Theorem ichv 48624

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

Assertion

Ref	Expression
ichv	⊢ [𝑥⇄𝑦]𝜑

Distinct variable groups:   𝜑,𝑥   𝜑,𝑦

Proof of Theorem ichv

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbv 2124	. . . . . . 7 ⊢ ([𝑎 / 𝑦]𝜑 ↔ 𝜑)
2	1	sbbii 2112	. . . . . 6 ⊢ ([𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)
3		sbv 2124	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)
4	2, 3	bitri 278	. . . . 5 ⊢ ([𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ 𝜑)
5	4	sbbii 2112	. . . 4 ⊢ ([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ [𝑥 / 𝑎]𝜑)
6		sbv 2124	. . . 4 ⊢ ([𝑥 / 𝑎]𝜑 ↔ 𝜑)
7	5, 6	bitri 278	. . 3 ⊢ ([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ 𝜑)
8	7	gen2 1828	. 2 ⊢ ∀𝑥∀𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ 𝜑)
9		df-ich 48621	. 2 ⊢ ([𝑥⇄𝑦]𝜑 ↔ ∀𝑥∀𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ 𝜑))
10	8, 9	mpbir 234	1 ⊢ [𝑥⇄𝑦]𝜑

This proof depends on syntax axioms:   ↔ wb 209  ∀wal 1567  [wsb 2098  [wich 48620

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-sb 2099  df-ich 48621

This theorem is used by: (None)