Theorem ichv 44789

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 2092	. . . . . . 7 ⊢ ([𝑎 / 𝑦]𝜑 ↔ 𝜑)
2	1	sbbii 2080	. . . . . 6 ⊢ ([𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)
3		sbv 2092	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)
4	2, 3	bitri 274	. . . . 5 ⊢ ([𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ 𝜑)
5	4	sbbii 2080	. . . 4 ⊢ ([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ [𝑥 / 𝑎]𝜑)
6		sbv 2092	. . . 4 ⊢ ([𝑥 / 𝑎]𝜑 ↔ 𝜑)
7	5, 6	bitri 274	. . 3 ⊢ ([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ 𝜑)
8	7	gen2 1800	. 2 ⊢ ∀𝑥∀𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ 𝜑)
9		df-ich 44786	. 2 ⊢ ([𝑥⇄𝑦]𝜑 ↔ ∀𝑥∀𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ 𝜑))
10	8, 9	mpbir 230	1 ⊢ [𝑥⇄𝑦]𝜑

Syntax hints:   ↔ wb 205  ∀wal 1537  [wsb 2068  [wich 44785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972

This theorem depends on definitions:  df-bi 206  df-ex 1784  df-sb 2069  df-ich 44786

This theorem is referenced by: (None)