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 43682
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 4543 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥](𝐵𝐶) ⊆ 𝐶𝐴 / 𝑥(𝐵𝐶) ⊆ 𝐴 / 𝑥𝐶))
2 csbima12 6107 . . . . 5 𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
32a1i 11 . . . 4 (𝐴𝑉𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
43sseq1d 4034 . . 3 (𝐴𝑉 → (𝐴 / 𝑥(𝐵𝐶) ⊆ 𝐴 / 𝑥𝐶 ↔ (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) ⊆ 𝐴 / 𝑥𝐶))
51, 4bitrd 279 . 2 (𝐴𝑉 → ([𝐴 / 𝑥](𝐵𝐶) ⊆ 𝐶 ↔ (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) ⊆ 𝐴 / 𝑥𝐶))
6 df-he 43676 . . 3 (𝐵 hereditary 𝐶 ↔ (𝐵𝐶) ⊆ 𝐶)
76sbcbii 3859 . 2 ([𝐴 / 𝑥]𝐵 hereditary 𝐶[𝐴 / 𝑥](𝐵𝐶) ⊆ 𝐶)
8 df-he 43676 . 2 (𝐴 / 𝑥𝐵 hereditary 𝐴 / 𝑥𝐶 ↔ (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) ⊆ 𝐴 / 𝑥𝐶)
95, 7, 83bitr4g 314 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵 hereditary 𝐶𝐴 / 𝑥𝐵 hereditary 𝐴 / 𝑥𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1537  wcel 2103  [wsbc 3798  csb 3915  wss 3970  cima 5702   hereditary whe 43675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5170  df-opab 5232  df-xp 5705  df-cnv 5707  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-he 43676
This theorem is referenced by:  frege77  43843
  Copyright terms: Public domain W3C validator