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Theorem sbcssg 3440
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 sbcalg 2931 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦(𝑦𝐵𝑦𝐶) ↔ ∀𝑦[𝐴 / 𝑥](𝑦𝐵𝑦𝐶)))
2 sbcimg 2920 . . . . 5 (𝐴𝑉 → ([𝐴 / 𝑥](𝑦𝐵𝑦𝐶) ↔ ([𝐴 / 𝑥]𝑦𝐵[𝐴 / 𝑥]𝑦𝐶)))
3 sbcel2g 2992 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝐵𝑦𝐴 / 𝑥𝐵))
4 sbcel2g 2992 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝐶𝑦𝐴 / 𝑥𝐶))
53, 4imbi12d 233 . . . . 5 (𝐴𝑉 → (([𝐴 / 𝑥]𝑦𝐵[𝐴 / 𝑥]𝑦𝐶) ↔ (𝑦𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))
62, 5bitrd 187 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥](𝑦𝐵𝑦𝐶) ↔ (𝑦𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))
76albidv 1778 . . 3 (𝐴𝑉 → (∀𝑦[𝐴 / 𝑥](𝑦𝐵𝑦𝐶) ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))
81, 7bitrd 187 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦(𝑦𝐵𝑦𝐶) ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))
9 dfss2 3054 . . 3 (𝐵𝐶 ↔ ∀𝑦(𝑦𝐵𝑦𝐶))
109sbcbii 2938 . 2 ([𝐴 / 𝑥]𝐵𝐶[𝐴 / 𝑥]𝑦(𝑦𝐵𝑦𝐶))
11 dfss2 3054 . 2 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶 ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))
128, 10, 113bitr4g 222 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1312  wcel 1463  [wsbc 2880  csb 2973  wss 3039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-sbc 2881  df-csb 2974  df-in 3045  df-ss 3052
This theorem is referenced by:  sbcrel  4593  sbcfg  5239
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