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Theorem trsbc 44560
Description: Formula-building inference rule for class substitution, substituting a class variable for the setvar variable of the transitivity predicate. trsbc 44560 is trsbcVD 44897 without virtual deductions and was automatically derived from trsbcVD 44897 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
trsbc (𝐴𝑉 → ([𝐴 / 𝑥]Tr 𝑥 ↔ Tr 𝐴))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem trsbc
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sbcal 3849 . . 3 ([𝐴 / 𝑥]𝑧𝑦((𝑧𝑦𝑦𝑥) → 𝑧𝑥) ↔ ∀𝑧[𝐴 / 𝑥]𝑦((𝑧𝑦𝑦𝑥) → 𝑧𝑥))
2 sbcal 3849 . . . . 5 ([𝐴 / 𝑥]𝑦((𝑧𝑦𝑦𝑥) → 𝑧𝑥) ↔ ∀𝑦[𝐴 / 𝑥]((𝑧𝑦𝑦𝑥) → 𝑧𝑥))
3 sbcim2g 44558 . . . . . . . 8 (𝐴𝑉 → ([𝐴 / 𝑥](𝑧𝑦 → (𝑦𝑥𝑧𝑥)) ↔ ([𝐴 / 𝑥]𝑧𝑦 → ([𝐴 / 𝑥]𝑦𝑥[𝐴 / 𝑥]𝑧𝑥))))
4 sbcg 3863 . . . . . . . . 9 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧𝑦𝑧𝑦))
5 sbcel2gv 3857 . . . . . . . . 9 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝑥𝑦𝐴))
6 sbcel2gv 3857 . . . . . . . . 9 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧𝑥𝑧𝐴))
7 imbi13 44540 . . . . . . . . 9 (([𝐴 / 𝑥]𝑧𝑦𝑧𝑦) → (([𝐴 / 𝑥]𝑦𝑥𝑦𝐴) → (([𝐴 / 𝑥]𝑧𝑥𝑧𝐴) → (([𝐴 / 𝑥]𝑧𝑦 → ([𝐴 / 𝑥]𝑦𝑥[𝐴 / 𝑥]𝑧𝑥)) ↔ (𝑧𝑦 → (𝑦𝐴𝑧𝐴))))))
84, 5, 6, 7syl3c 66 . . . . . . . 8 (𝐴𝑉 → (([𝐴 / 𝑥]𝑧𝑦 → ([𝐴 / 𝑥]𝑦𝑥[𝐴 / 𝑥]𝑧𝑥)) ↔ (𝑧𝑦 → (𝑦𝐴𝑧𝐴))))
93, 8bitrd 279 . . . . . . 7 (𝐴𝑉 → ([𝐴 / 𝑥](𝑧𝑦 → (𝑦𝑥𝑧𝑥)) ↔ (𝑧𝑦 → (𝑦𝐴𝑧𝐴))))
10 pm3.31 449 . . . . . . . . 9 ((𝑧𝑦 → (𝑦𝑥𝑧𝑥)) → ((𝑧𝑦𝑦𝑥) → 𝑧𝑥))
11 pm3.3 448 . . . . . . . . 9 (((𝑧𝑦𝑦𝑥) → 𝑧𝑥) → (𝑧𝑦 → (𝑦𝑥𝑧𝑥)))
1210, 11impbii 209 . . . . . . . 8 ((𝑧𝑦 → (𝑦𝑥𝑧𝑥)) ↔ ((𝑧𝑦𝑦𝑥) → 𝑧𝑥))
1312sbcbii 3846 . . . . . . 7 ([𝐴 / 𝑥](𝑧𝑦 → (𝑦𝑥𝑧𝑥)) ↔ [𝐴 / 𝑥]((𝑧𝑦𝑦𝑥) → 𝑧𝑥))
14 pm3.31 449 . . . . . . . 8 ((𝑧𝑦 → (𝑦𝐴𝑧𝐴)) → ((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
15 pm3.3 448 . . . . . . . 8 (((𝑧𝑦𝑦𝐴) → 𝑧𝐴) → (𝑧𝑦 → (𝑦𝐴𝑧𝐴)))
1614, 15impbii 209 . . . . . . 7 ((𝑧𝑦 → (𝑦𝐴𝑧𝐴)) ↔ ((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
179, 13, 163bitr3g 313 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥]((𝑧𝑦𝑦𝑥) → 𝑧𝑥) ↔ ((𝑧𝑦𝑦𝐴) → 𝑧𝐴)))
1817albidv 1920 . . . . 5 (𝐴𝑉 → (∀𝑦[𝐴 / 𝑥]((𝑧𝑦𝑦𝑥) → 𝑧𝑥) ↔ ∀𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴)))
192, 18bitrid 283 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦((𝑧𝑦𝑦𝑥) → 𝑧𝑥) ↔ ∀𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴)))
2019albidv 1920 . . 3 (𝐴𝑉 → (∀𝑧[𝐴 / 𝑥]𝑦((𝑧𝑦𝑦𝑥) → 𝑧𝑥) ↔ ∀𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴)))
211, 20bitrid 283 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧𝑦((𝑧𝑦𝑦𝑥) → 𝑧𝑥) ↔ ∀𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴)))
22 dftr2 5261 . . 3 (Tr 𝑥 ↔ ∀𝑧𝑦((𝑧𝑦𝑦𝑥) → 𝑧𝑥))
2322sbcbii 3846 . 2 ([𝐴 / 𝑥]Tr 𝑥[𝐴 / 𝑥]𝑧𝑦((𝑧𝑦𝑦𝑥) → 𝑧𝑥))
24 dftr2 5261 . 2 (Tr 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
2521, 23, 243bitr4g 314 1 (𝐴𝑉 → ([𝐴 / 𝑥]Tr 𝑥 ↔ Tr 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1538  wcel 2108  [wsbc 3788  Tr wtr 5259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-sbc 3789  df-ss 3968  df-uni 4908  df-tr 5260
This theorem is referenced by:  truniALT  44561  truniALTVD  44898  trintALTVD  44900  trintALT  44901
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