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Theorem csbieb 3921
Description: Bidirectional conversion between an implicit class substitution hypothesis 𝑥 = 𝐴𝐵 = 𝐶 and its explicit substitution equivalent. (Contributed by NM, 2-Mar-2008.)
Hypotheses
Ref Expression
csbieb.1 𝐴 ∈ V
csbieb.2 𝑥𝐶
Assertion
Ref Expression
csbieb (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem csbieb
StepHypRef Expression
1 csbieb.1 . 2 𝐴 ∈ V
2 csbieb.2 . 2 𝑥𝐶
3 csbiebt 3919 . 2 ((𝐴 ∈ V ∧ 𝑥𝐶) → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
41, 2, 3mp2an 690 1 (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1539   = wceq 1541  wcel 2106  wnfc 2882  Vcvv 3473  csb 3889
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 2702
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 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-v 3475  df-sbc 3774  df-csb 3890
This theorem is referenced by:  csbiebg  3922
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