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Theorem sbcssOLD 37777
 Description: Distribute proper substitution through a subclass relation. This theorem was automatically derived from sbcssgVD 38141. (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 3557 . . . 4 (𝐶𝐷 ↔ ∀𝑦(𝑦𝐶𝑦𝐷))
21sbcbiiOLD 37762 . . 3 (𝐴𝐵 → ([𝐴 / 𝑥]𝐶𝐷[𝐴 / 𝑥]𝑦(𝑦𝐶𝑦𝐷)))
3 sbcalgOLD 37773 . . . 4 (𝐴𝐵 → ([𝐴 / 𝑥]𝑦(𝑦𝐶𝑦𝐷) ↔ ∀𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷)))
4 sbcimg 3444 . . . . . . 7 (𝐴𝐵 → ([𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ ([𝐴 / 𝑥]𝑦𝐶[𝐴 / 𝑥]𝑦𝐷)))
5 sbcel2gOLD 37776 . . . . . . . 8 (𝐴𝐵 → ([𝐴 / 𝑥]𝑦𝐶𝑦𝐴 / 𝑥𝐶))
6 sbcel2gOLD 37776 . . . . . . . 8 (𝐴𝐵 → ([𝐴 / 𝑥]𝑦𝐷𝑦𝐴 / 𝑥𝐷))
75, 6imbi12d 333 . . . . . . 7 (𝐴𝐵 → (([𝐴 / 𝑥]𝑦𝐶[𝐴 / 𝑥]𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)))
84, 7bitrd 267 . . . . . 6 (𝐴𝐵 → ([𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)))
98alrimiv 1842 . . . . 5 (𝐴𝐵 → ∀𝑦([𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)))
10 albi 1736 . . . . 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 267 . . 3 (𝐴𝐵 → ([𝐴 / 𝑥]𝑦(𝑦𝐶𝑦𝐷) ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)))
132, 12bitrd 267 . 2 (𝐴𝐵 → ([𝐴 / 𝑥]𝐶𝐷 ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)))
14 dfss2 3557 . 2 (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷 ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))
1513, 14syl6bbr 277 1 (𝐴𝐵 → ([𝐴 / 𝑥]𝐶𝐷𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473   ∈ wcel 1977  [wsbc 3402  ⦋csb 3499   ⊆ wss 3540 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-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-sbc 3403  df-csb 3500  df-in 3547  df-ss 3554 This theorem is referenced by: (None)
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