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Theorem csbie2 3019
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 1411 . 2 𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷)
3 csbie2t.1 . . 3 𝐴 ∈ V
4 csbie2t.2 . . 3 𝐵 ∈ V
53, 4csbie2t 3018 . 2 (∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷) → 𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐷)
62, 5ax-mp 5 1 𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐷
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wal 1314   = wceq 1316  wcel 1465  Vcvv 2660  csb 2975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-sbc 2883  df-csb 2976
This theorem is referenced by:  fsumcnv  11174
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