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Theorem sbcssg 4118
Description: Distribute proper substitution through a subclass relation. (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Alexander van der Vekens, 23-Jul-2017.)
Assertion
Ref Expression
sbcssg (𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))

Proof of Theorem sbcssg
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 sbcal 3518 . . 3 ([𝐴 / 𝑥]𝑦(𝑦𝐵𝑦𝐶) ↔ ∀𝑦[𝐴 / 𝑥](𝑦𝐵𝑦𝐶))
2 sbcimg 3510 . . . . 5 (𝐴𝑉 → ([𝐴 / 𝑥](𝑦𝐵𝑦𝐶) ↔ ([𝐴 / 𝑥]𝑦𝐵[𝐴 / 𝑥]𝑦𝐶)))
3 sbcel2 4022 . . . . . 6 ([𝐴 / 𝑥]𝑦𝐵𝑦𝐴 / 𝑥𝐵)
4 sbcel2 4022 . . . . . 6 ([𝐴 / 𝑥]𝑦𝐶𝑦𝐴 / 𝑥𝐶)
53, 4imbi12i 339 . . . . 5 (([𝐴 / 𝑥]𝑦𝐵[𝐴 / 𝑥]𝑦𝐶) ↔ (𝑦𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))
62, 5syl6bb 276 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥](𝑦𝐵𝑦𝐶) ↔ (𝑦𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))
76albidv 1889 . . 3 (𝐴𝑉 → (∀𝑦[𝐴 / 𝑥](𝑦𝐵𝑦𝐶) ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))
81, 7syl5bb 272 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦(𝑦𝐵𝑦𝐶) ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))
9 dfss2 3624 . . 3 (𝐵𝐶 ↔ ∀𝑦(𝑦𝐵𝑦𝐶))
109sbcbii 3524 . 2 ([𝐴 / 𝑥]𝐵𝐶[𝐴 / 𝑥]𝑦(𝑦𝐵𝑦𝐶))
11 dfss2 3624 . 2 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶 ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))
128, 10, 113bitr4g 303 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1521  wcel 2030  [wsbc 3468  csb 3566  wss 3607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-in 3614  df-ss 3621  df-nul 3949
This theorem is referenced by:  sbcrel  5239  sbcfg  6081  iuninc  29505  csbwrecsg  33303  brtrclfv2  38336  cotrclrcl  38351  sbcheg  38390
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