Theorem ichcom 48361

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 2196	. . 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 2209	. . . . 5 ⊢ ([𝑏 / 𝑥][𝑎 / 𝑦]𝜓 ↔ [𝑎 / 𝑦][𝑏 / 𝑥]𝜓)
3		sbcom2 2209	. . . . 5 ⊢ ([𝑎 / 𝑥][𝑏 / 𝑦]𝜓 ↔ [𝑏 / 𝑦][𝑎 / 𝑥]𝜓)
4	2, 3	bibi12i 342	. . . 4 ⊢ (([𝑏 / 𝑥][𝑎 / 𝑦]𝜓 ↔ [𝑎 / 𝑥][𝑏 / 𝑦]𝜓) ↔ ([𝑎 / 𝑦][𝑏 / 𝑥]𝜓 ↔ [𝑏 / 𝑦][𝑎 / 𝑥]𝜓))
5	4	2albii 1853	. . 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 48360	. 2 ⊢ ([𝑥⇄𝑦]𝜓 ↔ ∀𝑏∀𝑎([𝑏 / 𝑥][𝑎 / 𝑦]𝜓 ↔ [𝑎 / 𝑥][𝑏 / 𝑦]𝜓))
8		dfich2 48360	. 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)

This proof depends on syntax axioms:   ↔ wb 209  ∀wal 1568  [wsb 2099  [wich 48347

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2178  ax-11 2194  ax-12 2215

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-ich 48348

This theorem is used by:  ichnfb  48367  ich2exprop  48373