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Theorem csbie2 3898
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 1799 . 2 𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷)
3 csbie2t.1 . . 3 𝐴 ∈ V
4 csbie2t.2 . . 3 𝐵 ∈ V
53, 4csbie2t 3897 . 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 397  wal 1540   = wceq 1542  wcel 2107  Vcvv 3444  csb 3856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-v 3446  df-sbc 3741  df-csb 3857
This theorem is referenced by:  fsumcnv  15663  fprodcnv  15871  dfrhm2  20155  mamufval  21750  mvmulfval  21907  vtxdgfval  28457  rnghmval  46275
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