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

Theorem sbcbig 3826
 Description: Distribution of class substitution over biconditional. (Contributed by Raph Levien, 10-Apr-2004.)
Assertion
Ref Expression
sbcbig (𝐴𝑉 → ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))

Proof of Theorem sbcbig
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 3778 . 2 (𝑦 = 𝐴 → ([𝑦 / 𝑥](𝜑𝜓) ↔ [𝐴 / 𝑥](𝜑𝜓)))
2 dfsbcq2 3778 . . 3 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
3 dfsbcq2 3778 . . 3 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜓[𝐴 / 𝑥]𝜓))
42, 3bibi12d 347 . 2 (𝑦 = 𝐴 → (([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
5 sbbi 2311 . 2 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓))
61, 4, 5vtoclbg 3573 1 (𝐴𝑉 → ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   = wceq 1530  [wsb 2062   ∈ wcel 2106  [wsbc 3775 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2167  ax-ext 2796 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2803  df-cleq 2817  df-clel 2897  df-sbc 3776 This theorem is referenced by:  sbcbi1  3833  sbcabel  3864  opsbc2ie  30154  bnj89  31878  bj-sbeq  34103  bj-sbceqgALT  34104  sbcbi  40735  sbc3orgVD  41047  sbcbiVD  41072
 Copyright terms: Public domain W3C validator