Theorem ichid 48123

Description: A setvar variable is always interchangeable with itself. (Contributed by AV, 29-Jul-2023.)

Assertion

Ref	Expression
ichid	⊢ [𝑥⇄𝑥]𝜑

Proof of Theorem ichid

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbid 2297	. . . . 5 ⊢ ([𝑥 / 𝑥][𝑎 / 𝑥]𝜑 ↔ [𝑎 / 𝑥]𝜑)
2	1	sbbii 2116	. . . 4 ⊢ ([𝑥 / 𝑎][𝑥 / 𝑥][𝑎 / 𝑥]𝜑 ↔ [𝑥 / 𝑎][𝑎 / 𝑥]𝜑)
3		sbid2vw 2301	. . . 4 ⊢ ([𝑥 / 𝑎][𝑎 / 𝑥]𝜑 ↔ 𝜑)
4	2, 3	bitri 278	. . 3 ⊢ ([𝑥 / 𝑎][𝑥 / 𝑥][𝑎 / 𝑥]𝜑 ↔ 𝜑)
5	4	gen2 1823	. 2 ⊢ ∀𝑥∀𝑥([𝑥 / 𝑎][𝑥 / 𝑥][𝑎 / 𝑥]𝜑 ↔ 𝜑)
6		df-ich 48118	. 2 ⊢ ([𝑥⇄𝑥]𝜑 ↔ ∀𝑥∀𝑥([𝑥 / 𝑎][𝑥 / 𝑥][𝑎 / 𝑥]𝜑 ↔ 𝜑))
7	5, 6	mpbir 234	1 ⊢ [𝑥⇄𝑥]𝜑

Syntax hints:   ↔ wb 209  ∀wal 1565  [wsb 2097  [wich 48117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-12 2219

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-sb 2098  df-ich 48118

This theorem is referenced by:  icheqid  48133