Theorem ichcom 44527

Description: The interchangeability of setvar variables is commutative. (Contributed by AV, 20-Aug-2023.)

Assertion

Ref	Expression
ichcom	⊢ ([𝑥⇄𝑦]𝜓 ↔ [𝑦⇄𝑥]𝜓)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜓(𝑥,𝑦)

Proof of Theorem ichcom

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		alcom 2162	. . 3 ⊢ (∀𝑏∀𝑎([𝑏 / 𝑥][𝑎 / 𝑦]𝜓 ↔ [𝑎 / 𝑥][𝑏 / 𝑦]𝜓) ↔ ∀𝑎∀𝑏([𝑏 / 𝑥][𝑎 / 𝑦]𝜓 ↔ [𝑎 / 𝑥][𝑏 / 𝑦]𝜓))
2		sbcom2 2167	. . . . 5 ⊢ ([𝑏 / 𝑥][𝑎 / 𝑦]𝜓 ↔ [𝑎 / 𝑦][𝑏 / 𝑥]𝜓)
3		sbcom2 2167	. . . . 5 ⊢ ([𝑎 / 𝑥][𝑏 / 𝑦]𝜓 ↔ [𝑏 / 𝑦][𝑎 / 𝑥]𝜓)
4	2, 3	bibi12i 343	. . . 4 ⊢ (([𝑏 / 𝑥][𝑎 / 𝑦]𝜓 ↔ [𝑎 / 𝑥][𝑏 / 𝑦]𝜓) ↔ ([𝑎 / 𝑦][𝑏 / 𝑥]𝜓 ↔ [𝑏 / 𝑦][𝑎 / 𝑥]𝜓))
5	4	2albii 1828	. . 3 ⊢ (∀𝑎∀𝑏([𝑏 / 𝑥][𝑎 / 𝑦]𝜓 ↔ [𝑎 / 𝑥][𝑏 / 𝑦]𝜓) ↔ ∀𝑎∀𝑏([𝑎 / 𝑦][𝑏 / 𝑥]𝜓 ↔ [𝑏 / 𝑦][𝑎 / 𝑥]𝜓))
6	1, 5	bitri 278	. 2 ⊢ (∀𝑏∀𝑎([𝑏 / 𝑥][𝑎 / 𝑦]𝜓 ↔ [𝑎 / 𝑥][𝑏 / 𝑦]𝜓) ↔ ∀𝑎∀𝑏([𝑎 / 𝑦][𝑏 / 𝑥]𝜓 ↔ [𝑏 / 𝑦][𝑎 / 𝑥]𝜓))
7		dfich2 44526	. 2 ⊢ ([𝑥⇄𝑦]𝜓 ↔ ∀𝑏∀𝑎([𝑏 / 𝑥][𝑎 / 𝑦]𝜓 ↔ [𝑎 / 𝑥][𝑏 / 𝑦]𝜓))
8		dfich2 44526	. 2 ⊢ ([𝑦⇄𝑥]𝜓 ↔ ∀𝑎∀𝑏([𝑎 / 𝑦][𝑏 / 𝑥]𝜓 ↔ [𝑏 / 𝑦][𝑎 / 𝑥]𝜓))
9	6, 7, 8	3bitr4i 306	1 ⊢ ([𝑥⇄𝑦]𝜓 ↔ [𝑦⇄𝑥]𝜓)

Syntax hints:   ↔ wb 209  ∀wal 1541  [wsb 2072  [wich 44513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-10 2143  ax-11 2160  ax-12 2177

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-nf 1792  df-sb 2073  df-ich 44514

This theorem is referenced by:  ichnfb  44533  ich2exprop  44539