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Theorem csbie2 3528
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 1713 . 2 𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷)
3 csbie2t.1 . . 3 𝐴 ∈ V
4 csbie2t.2 . . 3 𝐵 ∈ V
53, 4csbie2t 3527 . 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 382  wal 1472   = wceq 1474  wcel 1976  Vcvv 3172  csb 3498
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-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-v 3174  df-sbc 3402  df-csb 3499
This theorem is referenced by:  fsumcnv  14294  fprodcnv  14500  dfrhm2  18488  mamufval  19957  mvmulfval  20114  vtxdgfval  40664  rnghmval  41662
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