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Theorem sbcssOLD 38273
Description: Distribute proper substitution through a subclass relation. This theorem was automatically derived from sbcssgVD 38637. (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 dfss2 3576 . . . 4 (𝐶𝐷 ↔ ∀𝑦(𝑦𝐶𝑦𝐷))
21sbcbiiOLD 38258 . . 3 (𝐴𝐵 → ([𝐴 / 𝑥]𝐶𝐷[𝐴 / 𝑥]𝑦(𝑦𝐶𝑦𝐷)))
3 sbcalgOLD 38269 . . . 4 (𝐴𝐵 → ([𝐴 / 𝑥]𝑦(𝑦𝐶𝑦𝐷) ↔ ∀𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷)))
4 sbcimg 3463 . . . . . . 7 (𝐴𝐵 → ([𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ ([𝐴 / 𝑥]𝑦𝐶[𝐴 / 𝑥]𝑦𝐷)))
5 sbcel2gOLD 38272 . . . . . . . 8 (𝐴𝐵 → ([𝐴 / 𝑥]𝑦𝐶𝑦𝐴 / 𝑥𝐶))
6 sbcel2gOLD 38272 . . . . . . . 8 (𝐴𝐵 → ([𝐴 / 𝑥]𝑦𝐷𝑦𝐴 / 𝑥𝐷))
75, 6imbi12d 334 . . . . . . 7 (𝐴𝐵 → (([𝐴 / 𝑥]𝑦𝐶[𝐴 / 𝑥]𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)))
84, 7bitrd 268 . . . . . 6 (𝐴𝐵 → ([𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)))
98alrimiv 1852 . . . . 5 (𝐴𝐵 → ∀𝑦([𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)))
10 albi 1743 . . . . 5 (∀𝑦([𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)) → (∀𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)))
119, 10syl 17 . . . 4 (𝐴𝐵 → (∀𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)))
123, 11bitrd 268 . . 3 (𝐴𝐵 → ([𝐴 / 𝑥]𝑦(𝑦𝐶𝑦𝐷) ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)))
132, 12bitrd 268 . 2 (𝐴𝐵 → ([𝐴 / 𝑥]𝐶𝐷 ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)))
14 dfss2 3576 . 2 (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷 ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))
1513, 14syl6bbr 278 1 (𝐴𝐵 → ([𝐴 / 𝑥]𝐶𝐷𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1478  wcel 1987  [wsbc 3421  csb 3518  wss 3559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3191  df-sbc 3422  df-csb 3519  df-in 3566  df-ss 3573
This theorem is referenced by: (None)
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