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Theorem csbiebg 3860
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 2810 . . . 4 (𝑎 = 𝐴 → (𝑥 = 𝑎𝑥 = 𝐴))
21imbi1d 345 . . 3 (𝑎 = 𝐴 → ((𝑥 = 𝑎𝐵 = 𝐶) ↔ (𝑥 = 𝐴𝐵 = 𝐶)))
32albidv 1921 . 2 (𝑎 = 𝐴 → (∀𝑥(𝑥 = 𝑎𝐵 = 𝐶) ↔ ∀𝑥(𝑥 = 𝐴𝐵 = 𝐶)))
4 csbeq1 3831 . . 3 (𝑎 = 𝐴𝑎 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
54eqeq1d 2800 . 2 (𝑎 = 𝐴 → (𝑎 / 𝑥𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐶))
6 vex 3444 . . 3 𝑎 ∈ V
7 csbiebg.2 . . 3 𝑥𝐶
86, 7csbieb 3859 . 2 (∀𝑥(𝑥 = 𝑎𝐵 = 𝐶) ↔ 𝑎 / 𝑥𝐵 = 𝐶)
93, 5, 8vtoclbg 3517 1 (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1536   = wceq 1538  wcel 2111  wnfc 2936  csb 3828
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-v 3443  df-sbc 3721  df-csb 3829
This theorem is referenced by:  cdlemefrs29bpre0  37689
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