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Theorem csbie2 2895
 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 1339 . 2 𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷)
3 csbie2t.1 . . 3 𝐴 ∈ V
4 csbie2t.2 . . 3 𝐵 ∈ V
53, 4csbie2t 2894 . 2 (∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 = 𝐷) → 𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐷)
62, 5ax-mp 7 1 𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐷
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1241   = wceq 1243   ∈ wcel 1393  Vcvv 2557  ⦋csb 2852 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765  df-csb 2853 This theorem is referenced by: (None)
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