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Theorem sbcssOLD 39275
Description: Distribute proper substitution through a subclass relation. This theorem was automatically derived from sbcssgVD 39635. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sbcssOLD (𝐴𝐵 → ([𝐴 / 𝑥]𝐶𝐷𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷))

Proof of Theorem sbcssOLD
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 sbcal 3635 . . 3 ([𝐴 / 𝑥]𝑦(𝑦𝐶𝑦𝐷) ↔ ∀𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷))
2 sbcimg 3627 . . . . 5 (𝐴𝐵 → ([𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ ([𝐴 / 𝑥]𝑦𝐶[𝐴 / 𝑥]𝑦𝐷)))
3 sbcel2 4131 . . . . . . 7 ([𝐴 / 𝑥]𝑦𝐶𝑦𝐴 / 𝑥𝐶)
43a1i 11 . . . . . 6 (𝐴𝐵 → ([𝐴 / 𝑥]𝑦𝐶𝑦𝐴 / 𝑥𝐶))
5 sbcel2 4131 . . . . . . 7 ([𝐴 / 𝑥]𝑦𝐷𝑦𝐴 / 𝑥𝐷)
65a1i 11 . . . . . 6 (𝐴𝐵 → ([𝐴 / 𝑥]𝑦𝐷𝑦𝐴 / 𝑥𝐷))
74, 6imbi12d 333 . . . . 5 (𝐴𝐵 → (([𝐴 / 𝑥]𝑦𝐶[𝐴 / 𝑥]𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)))
82, 7bitrd 268 . . . 4 (𝐴𝐵 → ([𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)))
98albidv 2000 . . 3 (𝐴𝐵 → (∀𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)))
101, 9syl5bb 272 . 2 (𝐴𝐵 → ([𝐴 / 𝑥]𝑦(𝑦𝐶𝑦𝐷) ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)))
11 dfss2 3738 . . 3 (𝐶𝐷 ↔ ∀𝑦(𝑦𝐶𝑦𝐷))
1211sbcbii 3641 . 2 ([𝐴 / 𝑥]𝐶𝐷[𝐴 / 𝑥]𝑦(𝑦𝐶𝑦𝐷))
13 dfss2 3738 . 2 (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷 ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))
1410, 12, 133bitr4g 303 1 (𝐴𝐵 → ([𝐴 / 𝑥]𝐶𝐷𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1628  wcel 2144  [wsbc 3585  csb 3680  wss 3721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-in 3728  df-ss 3735  df-nul 4062
This theorem is referenced by: (None)
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