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Theorem sbcheg 37552
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 4057 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥](𝐵𝐶) ⊆ 𝐶𝐴 / 𝑥(𝐵𝐶) ⊆ 𝐴 / 𝑥𝐶))
2 csbima12 5442 . . . . 5 𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
32a1i 11 . . . 4 (𝐴𝑉𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
43sseq1d 3611 . . 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 37546 . . 3 (𝐵 hereditary 𝐶 ↔ (𝐵𝐶) ⊆ 𝐶)
76sbcbii 3473 . 2 ([𝐴 / 𝑥]𝐵 hereditary 𝐶[𝐴 / 𝑥](𝐵𝐶) ⊆ 𝐶)
8 df-he 37546 . 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 1480  wcel 1987  [wsbc 3417  csb 3514  wss 3555  cima 5077   hereditary whe 37545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-xp 5080  df-cnv 5082  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-he 37546
This theorem is referenced by:  frege77  37713
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