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Theorem csbiebg 2917
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 2065 . . . 4 (𝑎 = 𝐴 → (𝑥 = 𝑎𝑥 = 𝐴))
21imbi1d 224 . . 3 (𝑎 = 𝐴 → ((𝑥 = 𝑎𝐵 = 𝐶) ↔ (𝑥 = 𝐴𝐵 = 𝐶)))
32albidv 1721 . 2 (𝑎 = 𝐴 → (∀𝑥(𝑥 = 𝑎𝐵 = 𝐶) ↔ ∀𝑥(𝑥 = 𝐴𝐵 = 𝐶)))
4 csbeq1 2883 . . 3 (𝑎 = 𝐴𝑎 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
54eqeq1d 2064 . 2 (𝑎 = 𝐴 → (𝑎 / 𝑥𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐶))
6 vex 2577 . . 3 𝑎 ∈ V
7 csbiebg.2 . . 3 𝑥𝐶
86, 7csbieb 2916 . 2 (∀𝑥(𝑥 = 𝑎𝐵 = 𝐶) ↔ 𝑎 / 𝑥𝐵 = 𝐶)
93, 5, 8vtoclbg 2631 1 (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wal 1257   = wceq 1259  wcel 1409  wnfc 2181  csb 2880
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-sbc 2788  df-csb 2881
This theorem is referenced by: (None)
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