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Theorem sbcheg 37093
 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 4035 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥](𝐵𝐶) ⊆ 𝐶𝐴 / 𝑥(𝐵𝐶) ⊆ 𝐴 / 𝑥𝐶))
2 csbima12 5402 . . . . 5 𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
32a1i 11 . . . 4 (𝐴𝑉𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
43sseq1d 3595 . . 3 (𝐴𝑉 → (𝐴 / 𝑥(𝐵𝐶) ⊆ 𝐴 / 𝑥𝐶 ↔ (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) ⊆ 𝐴 / 𝑥𝐶))
51, 4bitrd 267 . 2 (𝐴𝑉 → ([𝐴 / 𝑥](𝐵𝐶) ⊆ 𝐶 ↔ (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) ⊆ 𝐴 / 𝑥𝐶))
6 df-he 37087 . . 3 (𝐵 hereditary 𝐶 ↔ (𝐵𝐶) ⊆ 𝐶)
76sbcbii 3458 . 2 ([𝐴 / 𝑥]𝐵 hereditary 𝐶[𝐴 / 𝑥](𝐵𝐶) ⊆ 𝐶)
8 df-he 37087 . 2 (𝐴 / 𝑥𝐵 hereditary 𝐴 / 𝑥𝐶 ↔ (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) ⊆ 𝐴 / 𝑥𝐶)
95, 7, 83bitr4g 302 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵 hereditary 𝐶𝐴 / 𝑥𝐵 hereditary 𝐴 / 𝑥𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977  [wsbc 3402  ⦋csb 3499   ⊆ wss 3540   “ cima 5041   hereditary whe 37086 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-he 37087 This theorem is referenced by:  frege77  37254
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