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Theorem trsbc 41016
 Description: Formula-building inference rule for class substitution, substituting a class variable for the setvar variable of the transitivity predicate. trsbc 41016 is trsbcVD 41353 without virtual deductions and was automatically derived from trsbcVD 41353 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 3811 . . 3 ([𝐴 / 𝑥]𝑧𝑦((𝑧𝑦𝑦𝑥) → 𝑧𝑥) ↔ ∀𝑧[𝐴 / 𝑥]𝑦((𝑧𝑦𝑦𝑥) → 𝑧𝑥))
2 sbcal 3811 . . . . 5 ([𝐴 / 𝑥]𝑦((𝑧𝑦𝑦𝑥) → 𝑧𝑥) ↔ ∀𝑦[𝐴 / 𝑥]((𝑧𝑦𝑦𝑥) → 𝑧𝑥))
3 sbcim2g 41014 . . . . . . . 8 (𝐴𝑉 → ([𝐴 / 𝑥](𝑧𝑦 → (𝑦𝑥𝑧𝑥)) ↔ ([𝐴 / 𝑥]𝑧𝑦 → ([𝐴 / 𝑥]𝑦𝑥[𝐴 / 𝑥]𝑧𝑥))))
4 sbcg 3824 . . . . . . . . 9 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧𝑦𝑧𝑦))
5 sbcel2gv 3818 . . . . . . . . 9 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝑥𝑦𝐴))
6 sbcel2gv 3818 . . . . . . . . 9 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧𝑥𝑧𝐴))
7 imbi13 40996 . . . . . . . . 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 281 . . . . . . 7 (𝐴𝑉 → ([𝐴 / 𝑥](𝑧𝑦 → (𝑦𝑥𝑧𝑥)) ↔ (𝑧𝑦 → (𝑦𝐴𝑧𝐴))))
10 pm3.31 452 . . . . . . . . 9 ((𝑧𝑦 → (𝑦𝑥𝑧𝑥)) → ((𝑧𝑦𝑦𝑥) → 𝑧𝑥))
11 pm3.3 451 . . . . . . . . 9 (((𝑧𝑦𝑦𝑥) → 𝑧𝑥) → (𝑧𝑦 → (𝑦𝑥𝑧𝑥)))
1210, 11impbii 211 . . . . . . . 8 ((𝑧𝑦 → (𝑦𝑥𝑧𝑥)) ↔ ((𝑧𝑦𝑦𝑥) → 𝑧𝑥))
1312sbcbii 3807 . . . . . . 7 ([𝐴 / 𝑥](𝑧𝑦 → (𝑦𝑥𝑧𝑥)) ↔ [𝐴 / 𝑥]((𝑧𝑦𝑦𝑥) → 𝑧𝑥))
14 pm3.31 452 . . . . . . . 8 ((𝑧𝑦 → (𝑦𝐴𝑧𝐴)) → ((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
15 pm3.3 451 . . . . . . . 8 (((𝑧𝑦𝑦𝐴) → 𝑧𝐴) → (𝑧𝑦 → (𝑦𝐴𝑧𝐴)))
1614, 15impbii 211 . . . . . . 7 ((𝑧𝑦 → (𝑦𝐴𝑧𝐴)) ↔ ((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
179, 13, 163bitr3g 315 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥]((𝑧𝑦𝑦𝑥) → 𝑧𝑥) ↔ ((𝑧𝑦𝑦𝐴) → 𝑧𝐴)))
1817albidv 1921 . . . . 5 (𝐴𝑉 → (∀𝑦[𝐴 / 𝑥]((𝑧𝑦𝑦𝑥) → 𝑧𝑥) ↔ ∀𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴)))
192, 18syl5bb 285 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦((𝑧𝑦𝑦𝑥) → 𝑧𝑥) ↔ ∀𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴)))
2019albidv 1921 . . 3 (𝐴𝑉 → (∀𝑧[𝐴 / 𝑥]𝑦((𝑧𝑦𝑦𝑥) → 𝑧𝑥) ↔ ∀𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴)))
211, 20syl5bb 285 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧𝑦((𝑧𝑦𝑦𝑥) → 𝑧𝑥) ↔ ∀𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴)))
22 dftr2 5148 . . 3 (Tr 𝑥 ↔ ∀𝑧𝑦((𝑧𝑦𝑦𝑥) → 𝑧𝑥))
2322sbcbii 3807 . 2 ([𝐴 / 𝑥]Tr 𝑥[𝐴 / 𝑥]𝑧𝑦((𝑧𝑦𝑦𝑥) → 𝑧𝑥))
24 dftr2 5148 . 2 (Tr 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
2521, 23, 243bitr4g 316 1 (𝐴𝑉 → ([𝐴 / 𝑥]Tr 𝑥 ↔ Tr 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1535   ∈ wcel 2114  [wsbc 3751  Tr wtr 5146 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-v 3475  df-sbc 3752  df-in 3919  df-ss 3928  df-uni 4813  df-tr 5147 This theorem is referenced by:  truniALT  41017  truniALTVD  41354  trintALTVD  41356  trintALT  41357
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