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

Theorem wl-sbrimt 33666
Description: Substitution with a variable not free in antecedent affects only the consequent. Closed form of sbrim 2543. (Contributed by Wolf Lammen, 26-Jul-2019.)
Assertion
Ref Expression
wl-sbrimt (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥](𝜑𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓)))

Proof of Theorem wl-sbrimt
StepHypRef Expression
1 sbim 2542 . 2 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
2 sbft 2526 . . 3 (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥]𝜑𝜑))
32imbi1d 330 . 2 (Ⅎ𝑥𝜑 → (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓)))
41, 3syl5bb 272 1 (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥](𝜑𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wnf 1856  [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 837  df-ex 1853  df-nf 1858  df-sb 2050
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator