Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sbcheg Structured version   Visualization version   GIF version

Theorem sbcheg 38567
Description: Distribute proper substitution through herditary relation. (Contributed by RP, 29-Jun-2020.)
Assertion
Ref Expression
sbcheg (𝐴𝑉 → ([𝐴 / 𝑥]𝐵 hereditary 𝐶𝐴 / 𝑥𝐵 hereditary 𝐴 / 𝑥𝐶))

Proof of Theorem sbcheg
StepHypRef Expression
1 sbcssg 4221 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥](𝐵𝐶) ⊆ 𝐶𝐴 / 𝑥(𝐵𝐶) ⊆ 𝐴 / 𝑥𝐶))
2 csbima12 5633 . . . . 5 𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
32a1i 11 . . . 4 (𝐴𝑉𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
43sseq1d 3765 . . 3 (𝐴𝑉 → (𝐴 / 𝑥(𝐵𝐶) ⊆ 𝐴 / 𝑥𝐶 ↔ (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) ⊆ 𝐴 / 𝑥𝐶))
51, 4bitrd 268 . 2 (𝐴𝑉 → ([𝐴 / 𝑥](𝐵𝐶) ⊆ 𝐶 ↔ (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) ⊆ 𝐴 / 𝑥𝐶))
6 df-he 38561 . . 3 (𝐵 hereditary 𝐶 ↔ (𝐵𝐶) ⊆ 𝐶)
76sbcbii 3624 . 2 ([𝐴 / 𝑥]𝐵 hereditary 𝐶[𝐴 / 𝑥](𝐵𝐶) ⊆ 𝐶)
8 df-he 38561 . 2 (𝐴 / 𝑥𝐵 hereditary 𝐴 / 𝑥𝐶 ↔ (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) ⊆ 𝐴 / 𝑥𝐶)
95, 7, 83bitr4g 303 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵 hereditary 𝐶𝐴 / 𝑥𝐵 hereditary 𝐴 / 𝑥𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1624  wcel 2131  [wsbc 3568  csb 3666  wss 3707  cima 5261   hereditary whe 38560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-fal 1630  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-br 4797  df-opab 4857  df-xp 5264  df-cnv 5266  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-he 38561
This theorem is referenced by:  frege77  38728
  Copyright terms: Public domain W3C validator