MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  csbiebg Structured version   Visualization version   GIF version

Theorem csbiebg 3705
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 2782 . . . 4 (𝑎 = 𝐴 → (𝑥 = 𝑎𝑥 = 𝐴))
21imbi1d 330 . . 3 (𝑎 = 𝐴 → ((𝑥 = 𝑎𝐵 = 𝐶) ↔ (𝑥 = 𝐴𝐵 = 𝐶)))
32albidv 2001 . 2 (𝑎 = 𝐴 → (∀𝑥(𝑥 = 𝑎𝐵 = 𝐶) ↔ ∀𝑥(𝑥 = 𝐴𝐵 = 𝐶)))
4 csbeq1 3685 . . 3 (𝑎 = 𝐴𝑎 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
54eqeq1d 2773 . 2 (𝑎 = 𝐴 → (𝑎 / 𝑥𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐶))
6 vex 3354 . . 3 𝑎 ∈ V
7 csbiebg.2 . . 3 𝑥𝐶
86, 7csbieb 3704 . 2 (∀𝑥(𝑥 = 𝑎𝐵 = 𝐶) ↔ 𝑎 / 𝑥𝐵 = 𝐶)
93, 5, 8vtoclbg 3418 1 (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ↔ 𝐴 / 𝑥𝐵 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1629   = wceq 1631  wcel 2145  wnfc 2900  csb 3682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-sbc 3588  df-csb 3683
This theorem is referenced by:  cdlemefrs29bpre0  36205
  Copyright terms: Public domain W3C validator