Users' Mathboxes Mathbox for Wolf Lammen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  wl-sb6rft Structured version   Visualization version   GIF version

Theorem wl-sb6rft 33682
Description: A specialization of wl-equsal1t 33679. Closed form of sb6rf 2573. (Contributed by Wolf Lammen, 27-Jul-2019.)
Assertion
Ref Expression
wl-sb6rft (Ⅎ𝑥𝜑 → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → [𝑥 / 𝑦]𝜑)))

Proof of Theorem wl-sb6rft
StepHypRef Expression
1 nfnf1 2190 . . 3 𝑥𝑥𝜑
2 id 22 . . 3 (Ⅎ𝑥𝜑 → Ⅎ𝑥𝜑)
3 sbequ12r 2271 . . . 4 (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑𝜑))
43a1i 11 . . 3 (Ⅎ𝑥𝜑 → (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑𝜑)))
51, 2, 4wl-equsald 33677 . 2 (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → [𝑥 / 𝑦]𝜑) ↔ 𝜑))
65bicomd 214 1 (Ⅎ𝑥𝜑 → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → [𝑥 / 𝑦]𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wal 1632  wnf 1859  [wsb 2052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-10 2177  ax-12 2206  ax-13 2411
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-ex 1856  df-nf 1861  df-sb 2053
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator