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

Theorem wl-sblimt 33003
Description: Substitution with a variable not free in antecedent affects only the consequent. Closed form of sbrim 2395. (Contributed by Wolf Lammen, 26-Jul-2019.)
Assertion
Ref Expression
wl-sblimt (Ⅎ𝑥𝜓 → ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑𝜓)))

Proof of Theorem wl-sblimt
StepHypRef Expression
1 sbim 2394 . 2 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
2 sbft 2378 . . 3 (Ⅎ𝑥𝜓 → ([𝑦 / 𝑥]𝜓𝜓))
32imbi2d 330 . 2 (Ⅎ𝑥𝜓 → (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ↔ ([𝑦 / 𝑥]𝜑𝜓)))
41, 3syl5bb 272 1 (Ⅎ𝑥𝜓 → ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wnf 1705  [wsb 1877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-12 2044  ax-13 2245
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1702  df-nf 1707  df-sb 1878
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator