Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ichnfb Structured version   Visualization version   GIF version

Theorem ichnfb 44605
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 44599 . . . 4 ([𝑥𝑦]𝜑 ↔ [𝑦𝑥]𝜑)
2 ichnfim 44604 . . . 4 ((∀𝑥𝑦𝜑 ∧ [𝑦𝑥]𝜑) → ∀𝑦𝑥𝜑)
31, 2sylan2b 597 . . 3 ((∀𝑥𝑦𝜑 ∧ [𝑥𝑦]𝜑) → ∀𝑦𝑥𝜑)
43expcom 417 . 2 ([𝑥𝑦]𝜑 → (∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑))
5 ichnfim 44604 . . 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 1541  wnf 1791  [wich 44585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-10 2142  ax-11 2159  ax-12 2176
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-nf 1792  df-sb 2072  df-ich 44586
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator