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Theorem sbcheg 40411
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 4435 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥](𝐵𝐶) ⊆ 𝐶𝐴 / 𝑥(𝐵𝐶) ⊆ 𝐴 / 𝑥𝐶))
2 csbima12 5925 . . . . 5 𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
32a1i 11 . . . 4 (𝐴𝑉𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
43sseq1d 3973 . . 3 (𝐴𝑉 → (𝐴 / 𝑥(𝐵𝐶) ⊆ 𝐴 / 𝑥𝐶 ↔ (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) ⊆ 𝐴 / 𝑥𝐶))
51, 4bitrd 282 . 2 (𝐴𝑉 → ([𝐴 / 𝑥](𝐵𝐶) ⊆ 𝐶 ↔ (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) ⊆ 𝐴 / 𝑥𝐶))
6 df-he 40405 . . 3 (𝐵 hereditary 𝐶 ↔ (𝐵𝐶) ⊆ 𝐶)
76sbcbii 3803 . 2 ([𝐴 / 𝑥]𝐵 hereditary 𝐶[𝐴 / 𝑥](𝐵𝐶) ⊆ 𝐶)
8 df-he 40405 . 2 (𝐴 / 𝑥𝐵 hereditary 𝐴 / 𝑥𝐶 ↔ (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) ⊆ 𝐴 / 𝑥𝐶)
95, 7, 83bitr4g 317 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵 hereditary 𝐶𝐴 / 𝑥𝐵 hereditary 𝐴 / 𝑥𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  wcel 2114  [wsbc 3747  csb 3855  wss 3908  cima 5535   hereditary whe 40404
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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-br 5043  df-opab 5105  df-xp 5538  df-cnv 5540  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-he 40405
This theorem is referenced by:  frege77  40572
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