Theorem ichcom 47384

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 2157	. . 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 2171	. . . . 5 ⊢ ([𝑏 / 𝑥][𝑎 / 𝑦]𝜓 ↔ [𝑎 / 𝑦][𝑏 / 𝑥]𝜓)
3		sbcom2 2171	. . . . 5 ⊢ ([𝑎 / 𝑥][𝑏 / 𝑦]𝜓 ↔ [𝑏 / 𝑦][𝑎 / 𝑥]𝜓)
4	2, 3	bibi12i 339	. . . 4 ⊢ (([𝑏 / 𝑥][𝑎 / 𝑦]𝜓 ↔ [𝑎 / 𝑥][𝑏 / 𝑦]𝜓) ↔ ([𝑎 / 𝑦][𝑏 / 𝑥]𝜓 ↔ [𝑏 / 𝑦][𝑎 / 𝑥]𝜓))
5	4	2albii 1817	. . 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 275	. 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 47383	. 2 ⊢ ([𝑥⇄𝑦]𝜓 ↔ ∀𝑏∀𝑎([𝑏 / 𝑥][𝑎 / 𝑦]𝜓 ↔ [𝑎 / 𝑥][𝑏 / 𝑦]𝜓))
8		dfich2 47383	. 2 ⊢ ([𝑦⇄𝑥]𝜓 ↔ ∀𝑎∀𝑏([𝑎 / 𝑦][𝑏 / 𝑥]𝜓 ↔ [𝑏 / 𝑦][𝑎 / 𝑥]𝜓))
9	6, 7, 8	3bitr4i 303	1 ⊢ ([𝑥⇄𝑦]𝜓 ↔ [𝑦⇄𝑥]𝜓)

Syntax hints:   ↔ wb 206  ∀wal 1535  [wsb 2062  [wich 47370

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-10 2139  ax-11 2155  ax-12 2175

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-ich 47371

This theorem is referenced by:  ichnfb  47390  ich2exprop  47396