Theorem icht 44856

Description: A theorem is interchangeable. (Contributed by SN, 4-May-2024.)

Hypothesis

Ref	Expression
icht.1	⊢ 𝜑

Assertion

Ref	Expression
icht	⊢ [𝑥⇄𝑦]𝜑

Proof of Theorem icht

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		icht.1	. . . . . . 7 ⊢ 𝜑
2	1	sbt 2072	. . . . . 6 ⊢ [𝑎 / 𝑦]𝜑
3	2	sbt 2072	. . . . 5 ⊢ [𝑦 / 𝑥][𝑎 / 𝑦]𝜑
4	3	sbt 2072	. . . 4 ⊢ [𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑
5	4, 1	2th 263	. . 3 ⊢ ([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ 𝜑)
6	5	gen2 1802	. 2 ⊢ ∀𝑥∀𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ 𝜑)
7		df-ich 44850	. 2 ⊢ ([𝑥⇄𝑦]𝜑 ↔ ∀𝑥∀𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ 𝜑))
8	6, 7	mpbir 230	1 ⊢ [𝑥⇄𝑦]𝜑

Syntax hints:   ↔ wb 205  ∀wal 1539  [wsb 2070  [wich 44849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801

This theorem depends on definitions:  df-bi 206  df-sb 2071  df-ich 44850

This theorem is referenced by: (None)