MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  csbie2 Structured version   Visualization version   GIF version

Theorem csbie2 3896
Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 27-Aug-2007.)
Hypotheses
Ref Expression
csbie2t.1 𝐴 ∈ V
csbie2t.2 𝐵 ∈ V
csbie2.3 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷)
Assertion
Ref Expression
csbie2 𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐷
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐷,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑦)

Proof of Theorem csbie2
StepHypRef Expression
1 csbie2.3 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷)
21gen2 1798 . 2 𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷)
3 csbie2t.1 . . 3 𝐴 ∈ V
4 csbie2t.2 . . 3 𝐵 ∈ V
53, 4csbie2t 3895 . 2 (∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷) → 𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐷)
62, 5ax-mp 5 1 𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐷
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1539   = wceq 1541  wcel 2106  Vcvv 3444  csb 3854
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2888  df-v 3446  df-sbc 3739  df-csb 3855
This theorem is referenced by:  fsumcnv  15655  fprodcnv  15863  dfrhm2  20144  mamufval  21730  mvmulfval  21887  vtxdgfval  28313  rnghmval  46159
  Copyright terms: Public domain W3C validator