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Theorem ichcom 43973
Description: The interchangeability of setvar variables is commutative. (Contributed by AV, 20-Aug-2023.)
Assertion
Ref Expression
ichcom ([𝑥𝑦]𝜓 ↔ [𝑦𝑥]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.
StepHypRef Expression
1 alcom 2161 . . 3 (∀𝑏𝑎([𝑏 / 𝑥][𝑎 / 𝑦]𝜓 ↔ [𝑎 / 𝑥][𝑏 / 𝑦]𝜓) ↔ ∀𝑎𝑏([𝑏 / 𝑥][𝑎 / 𝑦]𝜓 ↔ [𝑎 / 𝑥][𝑏 / 𝑦]𝜓))
2 sbcom2 2166 . . . . 5 ([𝑏 / 𝑥][𝑎 / 𝑦]𝜓 ↔ [𝑎 / 𝑦][𝑏 / 𝑥]𝜓)
3 sbcom2 2166 . . . . 5 ([𝑎 / 𝑥][𝑏 / 𝑦]𝜓 ↔ [𝑏 / 𝑦][𝑎 / 𝑥]𝜓)
42, 3bibi12i 343 . . . 4 (([𝑏 / 𝑥][𝑎 / 𝑦]𝜓 ↔ [𝑎 / 𝑥][𝑏 / 𝑦]𝜓) ↔ ([𝑎 / 𝑦][𝑏 / 𝑥]𝜓 ↔ [𝑏 / 𝑦][𝑎 / 𝑥]𝜓))
542albii 1822 . . 3 (∀𝑎𝑏([𝑏 / 𝑥][𝑎 / 𝑦]𝜓 ↔ [𝑎 / 𝑥][𝑏 / 𝑦]𝜓) ↔ ∀𝑎𝑏([𝑎 / 𝑦][𝑏 / 𝑥]𝜓 ↔ [𝑏 / 𝑦][𝑎 / 𝑥]𝜓))
61, 5bitri 278 . 2 (∀𝑏𝑎([𝑏 / 𝑥][𝑎 / 𝑦]𝜓 ↔ [𝑎 / 𝑥][𝑏 / 𝑦]𝜓) ↔ ∀𝑎𝑏([𝑎 / 𝑦][𝑏 / 𝑥]𝜓 ↔ [𝑏 / 𝑦][𝑎 / 𝑥]𝜓))
7 dfich2 43972 . 2 ([𝑥𝑦]𝜓 ↔ ∀𝑏𝑎([𝑏 / 𝑥][𝑎 / 𝑦]𝜓 ↔ [𝑎 / 𝑥][𝑏 / 𝑦]𝜓))
8 dfich2 43972 . 2 ([𝑦𝑥]𝜓 ↔ ∀𝑎𝑏([𝑎 / 𝑦][𝑏 / 𝑥]𝜓 ↔ [𝑏 / 𝑦][𝑎 / 𝑥]𝜓))
96, 7, 83bitr4i 306 1 ([𝑥𝑦]𝜓 ↔ [𝑦𝑥]𝜓)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wal 1536  [wsb 2069  [wich 43959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2143  ax-11 2159  ax-12 2176
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-ich 43960
This theorem is referenced by:  ichnfb  43979  ich2exprop  43985
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