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Theorem ichnfim 43615
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 2152 . . . 4 𝑥𝑥𝜑
21nfal 2336 . . 3 𝑥𝑦𝑥𝜑
3 nfich1 43598 . . 3 𝑥[𝑥𝑦]𝜑
42, 3nfan 1894 . 2 𝑥(∀𝑦𝑥𝜑 ∧ [𝑥𝑦]𝜑)
5 dfich2 43604 . . . . 5 ([𝑥𝑦]𝜑 ↔ ∀𝑎𝑏([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑥][𝑎 / 𝑦]𝜑))
6 ichnfimlem3 43614 . . . . . . . 8 (∀𝑦𝑥𝜑 → ([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑))
7 ichnfimlem3 43614 . . . . . . . 8 (∀𝑦𝑥𝜑 → ([𝑏 / 𝑥][𝑎 / 𝑦]𝜑 ↔ [𝑎 / 𝑦]𝜑))
86, 7bibi12d 348 . . . . . . 7 (∀𝑦𝑥𝜑 → (([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑥][𝑎 / 𝑦]𝜑) ↔ ([𝑏 / 𝑦]𝜑 ↔ [𝑎 / 𝑦]𝜑)))
9 bicom1 223 . . . . . . 7 (([𝑏 / 𝑦]𝜑 ↔ [𝑎 / 𝑦]𝜑) → ([𝑎 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑))
108, 9syl6bi 255 . . . . . 6 (∀𝑦𝑥𝜑 → (([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑥][𝑎 / 𝑦]𝜑) → ([𝑎 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑)))
11102alimdv 1913 . . . . 5 (∀𝑦𝑥𝜑 → (∀𝑎𝑏([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑥][𝑎 / 𝑦]𝜑) → ∀𝑎𝑏([𝑎 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑)))
125, 11syl5bi 244 . . . 4 (∀𝑦𝑥𝜑 → ([𝑥𝑦]𝜑 → ∀𝑎𝑏([𝑎 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑)))
1312imp 409 . . 3 ((∀𝑦𝑥𝜑 ∧ [𝑥𝑦]𝜑) → ∀𝑎𝑏([𝑎 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑))
14 sbnf2 2371 . . 3 (Ⅎ𝑦𝜑 ↔ ∀𝑎𝑏([𝑎 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑))
1513, 14sylibr 236 . 2 ((∀𝑦𝑥𝜑 ∧ [𝑥𝑦]𝜑) → Ⅎ𝑦𝜑)
164, 15alrimi 2206 1 ((∀𝑦𝑥𝜑 ∧ [𝑥𝑦]𝜑) → ∀𝑥𝑦𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wal 1529  wnf 1778  [wsb 2063  [wich 43596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-10 2139  ax-11 2154  ax-12 2170  ax-13 2384
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-ich 43597
This theorem is referenced by:  ichnfb  43616
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