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Theorem csbiebg 3046
Description: Bidirectional conversion between an implicit class substitution hypothesis 𝑥 = 𝐴𝐵 = 𝐶 and its explicit substitution equivalent. (Contributed by NM, 24-Mar-2013.) (Revised by Mario Carneiro, 11-Dec-2016.)
Hypothesis
Ref Expression
csbiebg.2 𝑥𝐶
Assertion
Ref Expression
csbiebg (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)   𝑉(𝑥)

Proof of Theorem csbiebg
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 eqeq2 2150 . . . 4 (𝑎 = 𝐴 → (𝑥 = 𝑎𝑥 = 𝐴))
21imbi1d 230 . . 3 (𝑎 = 𝐴 → ((𝑥 = 𝑎𝐵 = 𝐶) ↔ (𝑥 = 𝐴𝐵 = 𝐶)))
32albidv 1797 . 2 (𝑎 = 𝐴 → (∀𝑥(𝑥 = 𝑎𝐵 = 𝐶) ↔ ∀𝑥(𝑥 = 𝐴𝐵 = 𝐶)))
4 csbeq1 3009 . . 3 (𝑎 = 𝐴𝑎 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
54eqeq1d 2149 . 2 (𝑎 = 𝐴 → (𝑎 / 𝑥𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐶))
6 vex 2692 . . 3 𝑎 ∈ V
7 csbiebg.2 . . 3 𝑥𝐶
86, 7csbieb 3045 . 2 (∀𝑥(𝑥 = 𝑎𝐵 = 𝐶) ↔ 𝑎 / 𝑥𝐵 = 𝐶)
93, 5, 8vtoclbg 2750 1 (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1330   = wceq 1332  wcel 1481  wnfc 2269  csb 3006
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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-sbc 2913  df-csb 3007
This theorem is referenced by: (None)
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