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Theorem ichnfim 44349
 Description: If in an interchangeability context 𝑥 is not free in 𝜑, the same holds for 𝑦. (Contributed by Wolf Lammen, 6-Aug-2023.) (Revised by AV, 23-Sep-2023.)
Assertion
Ref Expression
ichnfim ((∀𝑦𝑥𝜑 ∧ [𝑥𝑦]𝜑) → ∀𝑥𝑦𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem ichnfim
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfnf1 2155 . . . 4 𝑥𝑥𝜑
21nfal 2331 . . 3 𝑥𝑦𝑥𝜑
3 nfich1 44332 . . 3 𝑥[𝑥𝑦]𝜑
42, 3nfan 1900 . 2 𝑥(∀𝑦𝑥𝜑 ∧ [𝑥𝑦]𝜑)
5 dfich2 44343 . . . . 5 ([𝑥𝑦]𝜑 ↔ ∀𝑎𝑏([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑥][𝑎 / 𝑦]𝜑))
6 ichnfimlem 44348 . . . . . . . 8 (∀𝑦𝑥𝜑 → ([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑))
7 ichnfimlem 44348 . . . . . . . 8 (∀𝑦𝑥𝜑 → ([𝑏 / 𝑥][𝑎 / 𝑦]𝜑 ↔ [𝑎 / 𝑦]𝜑))
86, 7bibi12d 349 . . . . . . 7 (∀𝑦𝑥𝜑 → (([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑥][𝑎 / 𝑦]𝜑) ↔ ([𝑏 / 𝑦]𝜑 ↔ [𝑎 / 𝑦]𝜑)))
9 bicom1 224 . . . . . . 7 (([𝑏 / 𝑦]𝜑 ↔ [𝑎 / 𝑦]𝜑) → ([𝑎 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑))
108, 9syl6bi 256 . . . . . 6 (∀𝑦𝑥𝜑 → (([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑥][𝑎 / 𝑦]𝜑) → ([𝑎 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑)))
11102alimdv 1919 . . . . 5 (∀𝑦𝑥𝜑 → (∀𝑎𝑏([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑥][𝑎 / 𝑦]𝜑) → ∀𝑎𝑏([𝑎 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑)))
125, 11syl5bi 245 . . . 4 (∀𝑦𝑥𝜑 → ([𝑥𝑦]𝜑 → ∀𝑎𝑏([𝑎 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑)))
1312imp 410 . . 3 ((∀𝑦𝑥𝜑 ∧ [𝑥𝑦]𝜑) → ∀𝑎𝑏([𝑎 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑))
14 sbnf2 2366 . . 3 (Ⅎ𝑦𝜑 ↔ ∀𝑎𝑏([𝑎 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑))
1513, 14sylibr 237 . 2 ((∀𝑦𝑥𝜑 ∧ [𝑥𝑦]𝜑) → Ⅎ𝑦𝜑)
164, 15alrimi 2211 1 ((∀𝑦𝑥𝜑 ∧ [𝑥𝑦]𝜑) → ∀𝑥𝑦𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536  Ⅎwnf 1785  [wsb 2069  [wich 44330 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 2142  ax-11 2158  ax-12 2175 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 44331 This theorem is referenced by:  ichnfb  44350
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