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Theorem ichn 43960
Description: Negation does not affect interchangeability. (Contributed by SN, 30-Aug-2023.)
Assertion
Ref Expression
ichn ([𝑎𝑏]𝜑 ↔ [𝑎𝑏] ¬ 𝜑)

Proof of Theorem ichn
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 notbi 322 . . . 4 (([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑𝜑) ↔ (¬ [𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑 ↔ ¬ 𝜑))
2 sbn 2285 . . . . . . . . 9 ([𝑢 / 𝑏] ¬ 𝜑 ↔ ¬ [𝑢 / 𝑏]𝜑)
32sbbii 2081 . . . . . . . 8 ([𝑏 / 𝑎][𝑢 / 𝑏] ¬ 𝜑 ↔ [𝑏 / 𝑎] ¬ [𝑢 / 𝑏]𝜑)
4 sbn 2285 . . . . . . . 8 ([𝑏 / 𝑎] ¬ [𝑢 / 𝑏]𝜑 ↔ ¬ [𝑏 / 𝑎][𝑢 / 𝑏]𝜑)
53, 4bitri 278 . . . . . . 7 ([𝑏 / 𝑎][𝑢 / 𝑏] ¬ 𝜑 ↔ ¬ [𝑏 / 𝑎][𝑢 / 𝑏]𝜑)
65sbbii 2081 . . . . . 6 ([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏] ¬ 𝜑 ↔ [𝑎 / 𝑢] ¬ [𝑏 / 𝑎][𝑢 / 𝑏]𝜑)
7 sbn 2285 . . . . . 6 ([𝑎 / 𝑢] ¬ [𝑏 / 𝑎][𝑢 / 𝑏]𝜑 ↔ ¬ [𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑)
86, 7bitri 278 . . . . 5 ([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏] ¬ 𝜑 ↔ ¬ [𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑)
98bibi1i 342 . . . 4 (([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏] ¬ 𝜑 ↔ ¬ 𝜑) ↔ (¬ [𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑 ↔ ¬ 𝜑))
101, 9bitr4i 281 . . 3 (([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑𝜑) ↔ ([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏] ¬ 𝜑 ↔ ¬ 𝜑))
11102albii 1822 . 2 (∀𝑎𝑏([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑𝜑) ↔ ∀𝑎𝑏([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏] ¬ 𝜑 ↔ ¬ 𝜑))
12 df-ich 43950 . 2 ([𝑎𝑏]𝜑 ↔ ∀𝑎𝑏([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑𝜑))
13 df-ich 43950 . 2 ([𝑎𝑏] ¬ 𝜑 ↔ ∀𝑎𝑏([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏] ¬ 𝜑 ↔ ¬ 𝜑))
1411, 12, 133bitr4i 306 1 ([𝑎𝑏]𝜑 ↔ [𝑎𝑏] ¬ 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 209  wal 1536  [wsb 2069  [wich 43949
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-12 2176
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070  df-ich 43950
This theorem is referenced by:  ichim  43961
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