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Theorem csbiebg 2883
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 (A 𝑉 → (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 2046 . . . 4 (𝑎 = A → (x = 𝑎x = A))
21imbi1d 220 . . 3 (𝑎 = A → ((x = 𝑎B = 𝐶) ↔ (x = AB = 𝐶)))
32albidv 1702 . 2 (𝑎 = A → (x(x = 𝑎B = 𝐶) ↔ x(x = AB = 𝐶)))
4 csbeq1 2849 . . 3 (𝑎 = A𝑎 / xB = A / xB)
54eqeq1d 2045 . 2 (𝑎 = A → (𝑎 / xB = 𝐶A / xB = 𝐶))
6 vex 2554 . . 3 𝑎 V
7 csbiebg.2 . . 3 x𝐶
86, 7csbieb 2882 . 2 (x(x = 𝑎B = 𝐶) ↔ 𝑎 / xB = 𝐶)
93, 5, 8vtoclbg 2608 1 (A 𝑉 → (x(x = AB = 𝐶) ↔ A / xB = 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1240   = wceq 1242   wcel 1390  wnfc 2162  csb 2846
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sbc 2759  df-csb 2847
This theorem is referenced by: (None)
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