MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  spsbbi Structured version   Visualization version   GIF version

Theorem spsbbi 2390
Description: Specialization of biconditional. (Contributed by NM, 2-Jun-1993.)
Assertion
Ref Expression
spsbbi (∀𝑥(𝜑𝜓) → ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓))

Proof of Theorem spsbbi
StepHypRef Expression
1 stdpc4 2341 . 2 (∀𝑥(𝜑𝜓) → [𝑦 / 𝑥](𝜑𝜓))
2 sbbi 2389 . 2 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓))
31, 2sylib 207 1 (∀𝑥(𝜑𝜓) → ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wal 1473  [wsb 1867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034  ax-13 2234
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868
This theorem is referenced by:  sbbid  2391  sbeqi  33138
  Copyright terms: Public domain W3C validator