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Theorem ichnfb 43908
 Description: If 𝑥 and 𝑦 are interchangeable in 𝜑, they are both free or both not free in 𝜑. (Contributed by Wolf Lammen, 6-Aug-2023.) (Revised by AV, 23-Sep-2023.)
Assertion
Ref Expression
ichnfb ([𝑥𝑦]𝜑 → (∀𝑥𝑦𝜑 ↔ ∀𝑦𝑥𝜑))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem ichnfb
StepHypRef Expression
1 ichcom 43902 . . . 4 ([𝑥𝑦]𝜑 ↔ [𝑦𝑥]𝜑)
2 ichnfim 43907 . . . 4 ((∀𝑥𝑦𝜑 ∧ [𝑦𝑥]𝜑) → ∀𝑦𝑥𝜑)
31, 2sylan2b 596 . . 3 ((∀𝑥𝑦𝜑 ∧ [𝑥𝑦]𝜑) → ∀𝑦𝑥𝜑)
43expcom 417 . 2 ([𝑥𝑦]𝜑 → (∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑))
5 ichnfim 43907 . . 3 ((∀𝑦𝑥𝜑 ∧ [𝑥𝑦]𝜑) → ∀𝑥𝑦𝜑)
65expcom 417 . 2 ([𝑥𝑦]𝜑 → (∀𝑦𝑥𝜑 → ∀𝑥𝑦𝜑))
74, 6impbid 215 1 ([𝑥𝑦]𝜑 → (∀𝑥𝑦𝜑 ↔ ∀𝑦𝑥𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536  Ⅎwnf 1785  [wich 43888 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 1912  ax-6 1971  ax-7 2016  ax-10 2146  ax-11 2162  ax-12 2179 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 2071  df-ich 43889 This theorem is referenced by: (None)
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