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Theorem trsbc 38271
Description: Formula-building inference rule for class substitution, substituting a class variable for the setvar variable of the transitivity predicate. trsbc 38271 is trsbcVD 38635 without virtual deductions and was automatically derived from trsbcVD 38635 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 3472 . . 3 ([𝐴 / 𝑥]𝑧𝑦((𝑧𝑦𝑦𝑥) → 𝑧𝑥) ↔ ∀𝑧[𝐴 / 𝑥]𝑦((𝑧𝑦𝑦𝑥) → 𝑧𝑥))
2 sbcal 3472 . . . . 5 ([𝐴 / 𝑥]𝑦((𝑧𝑦𝑦𝑥) → 𝑧𝑥) ↔ ∀𝑦[𝐴 / 𝑥]((𝑧𝑦𝑦𝑥) → 𝑧𝑥))
3 sbcim2g 38269 . . . . . . . 8 (𝐴𝑉 → ([𝐴 / 𝑥](𝑧𝑦 → (𝑦𝑥𝑧𝑥)) ↔ ([𝐴 / 𝑥]𝑧𝑦 → ([𝐴 / 𝑥]𝑦𝑥[𝐴 / 𝑥]𝑧𝑥))))
4 sbcg 3490 . . . . . . . . 9 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧𝑦𝑧𝑦))
5 sbcel2gv 3483 . . . . . . . . 9 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝑥𝑦𝐴))
6 sbcel2gv 3483 . . . . . . . . 9 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧𝑥𝑧𝐴))
7 imbi13 38247 . . . . . . . . 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 268 . . . . . . 7 (𝐴𝑉 → ([𝐴 / 𝑥](𝑧𝑦 → (𝑦𝑥𝑧𝑥)) ↔ (𝑧𝑦 → (𝑦𝐴𝑧𝐴))))
10 pm3.31 461 . . . . . . . . 9 ((𝑧𝑦 → (𝑦𝑥𝑧𝑥)) → ((𝑧𝑦𝑦𝑥) → 𝑧𝑥))
11 pm3.3 460 . . . . . . . . 9 (((𝑧𝑦𝑦𝑥) → 𝑧𝑥) → (𝑧𝑦 → (𝑦𝑥𝑧𝑥)))
1210, 11impbii 199 . . . . . . . 8 ((𝑧𝑦 → (𝑦𝑥𝑧𝑥)) ↔ ((𝑧𝑦𝑦𝑥) → 𝑧𝑥))
1312sbcbii 3478 . . . . . . 7 ([𝐴 / 𝑥](𝑧𝑦 → (𝑦𝑥𝑧𝑥)) ↔ [𝐴 / 𝑥]((𝑧𝑦𝑦𝑥) → 𝑧𝑥))
14 pm3.31 461 . . . . . . . 8 ((𝑧𝑦 → (𝑦𝐴𝑧𝐴)) → ((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
15 pm3.3 460 . . . . . . . 8 (((𝑧𝑦𝑦𝐴) → 𝑧𝐴) → (𝑧𝑦 → (𝑦𝐴𝑧𝐴)))
1614, 15impbii 199 . . . . . . 7 ((𝑧𝑦 → (𝑦𝐴𝑧𝐴)) ↔ ((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
179, 13, 163bitr3g 302 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥]((𝑧𝑦𝑦𝑥) → 𝑧𝑥) ↔ ((𝑧𝑦𝑦𝐴) → 𝑧𝐴)))
1817albidv 1846 . . . . 5 (𝐴𝑉 → (∀𝑦[𝐴 / 𝑥]((𝑧𝑦𝑦𝑥) → 𝑧𝑥) ↔ ∀𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴)))
192, 18syl5bb 272 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦((𝑧𝑦𝑦𝑥) → 𝑧𝑥) ↔ ∀𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴)))
2019albidv 1846 . . 3 (𝐴𝑉 → (∀𝑧[𝐴 / 𝑥]𝑦((𝑧𝑦𝑦𝑥) → 𝑧𝑥) ↔ ∀𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴)))
211, 20syl5bb 272 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧𝑦((𝑧𝑦𝑦𝑥) → 𝑧𝑥) ↔ ∀𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴)))
22 dftr2 4724 . . 3 (Tr 𝑥 ↔ ∀𝑧𝑦((𝑧𝑦𝑦𝑥) → 𝑧𝑥))
2322sbcbii 3478 . 2 ([𝐴 / 𝑥]Tr 𝑥[𝐴 / 𝑥]𝑧𝑦((𝑧𝑦𝑦𝑥) → 𝑧𝑥))
24 dftr2 4724 . 2 (Tr 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
2521, 23, 243bitr4g 303 1 (𝐴𝑉 → ([𝐴 / 𝑥]Tr 𝑥 ↔ Tr 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1478  wcel 1987  [wsbc 3422  Tr wtr 4722
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 3192  df-sbc 3423  df-in 3567  df-ss 3574  df-uni 4410  df-tr 4723
This theorem is referenced by:  truniALT  38272  truniALTVD  38636  trintALTVD  38638  trintALT  38639
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