Users' Mathboxes Mathbox for Wolf Lammen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  wl-sbcom2d-lem1 Structured version   Visualization version   GIF version

Theorem wl-sbcom2d-lem1 33676
Description: Lemma used to prove wl-sbcom2d 33678. (Contributed by Wolf Lammen, 10-Aug-2019.) (New usage is discouraged.)
Assertion
Ref Expression
wl-sbcom2d-lem1 ((𝑢 = 𝑦𝑣 = 𝑤) → (¬ ∀𝑥 𝑥 = 𝑤 → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)))
Distinct variable groups:   𝑣,𝑢,𝑥   𝑦,𝑢,𝑣   𝑤,𝑢,𝑣   𝑧,𝑢,𝑣   𝜑,𝑢,𝑣
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem wl-sbcom2d-lem1
StepHypRef Expression
1 nfna1 2185 . . . . . 6 𝑥 ¬ ∀𝑥 𝑥 = 𝑤
2 nfeqf2 2452 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑤 → Ⅎ𝑥 𝑣 = 𝑤)
31, 2nfan1 2222 . . . . 5 𝑥(¬ ∀𝑥 𝑥 = 𝑤𝑣 = 𝑤)
4 sbequ 2523 . . . . . 6 (𝑣 = 𝑤 → ([𝑣 / 𝑧]𝜑 ↔ [𝑤 / 𝑧]𝜑))
54adantl 467 . . . . 5 ((¬ ∀𝑥 𝑥 = 𝑤𝑣 = 𝑤) → ([𝑣 / 𝑧]𝜑 ↔ [𝑤 / 𝑧]𝜑))
63, 5sbbid 2550 . . . 4 ((¬ ∀𝑥 𝑥 = 𝑤𝑣 = 𝑤) → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑢 / 𝑥][𝑤 / 𝑧]𝜑))
76ancoms 455 . . 3 ((𝑣 = 𝑤 ∧ ¬ ∀𝑥 𝑥 = 𝑤) → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑢 / 𝑥][𝑤 / 𝑧]𝜑))
8 sbequ 2523 . . 3 (𝑢 = 𝑦 → ([𝑢 / 𝑥][𝑤 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑))
97, 8sylan9bbr 500 . 2 ((𝑢 = 𝑦 ∧ (𝑣 = 𝑤 ∧ ¬ ∀𝑥 𝑥 = 𝑤)) → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑))
109expr 444 1 ((𝑢 = 𝑦𝑣 = 𝑤) → (¬ ∀𝑥 𝑥 = 𝑤 → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wal 1629  [wsb 2049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-10 2174  ax-12 2203  ax-13 2408
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-ex 1853  df-nf 1858  df-sb 2050
This theorem is referenced by:  wl-sbcom2d  33678
  Copyright terms: Public domain W3C validator