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

Theorem spsbbi 2389
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 2340 . 2 (∀𝑥(𝜑𝜓) → [𝑦 / 𝑥](𝜑𝜓))
2 sbbi 2388 . 2 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓))
31, 2sylib 206 1 (∀𝑥(𝜑𝜓) → ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wal 1472  [wsb 1866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-12 2033  ax-13 2233
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867
This theorem is referenced by:  sbbid  2390  sbeqi  32934
  Copyright terms: Public domain W3C validator