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Theorem dftr5 3848
Description: An alternate way of defining a transitive class. (Contributed by NM, 20-Mar-2004.)
Assertion
Ref Expression
dftr5 (Tr Ax A y x y A)
Distinct variable group:   x,y,A

Proof of Theorem dftr5
StepHypRef Expression
1 dftr2 3847 . 2 (Tr Ayx((y x x A) → y A))
2 alcom 1364 . . 3 (yx((y x x A) → y A) ↔ xy((y x x A) → y A))
3 impexp 250 . . . . . . . 8 (((y x x A) → y A) ↔ (y x → (x Ay A)))
43albii 1356 . . . . . . 7 (y((y x x A) → y A) ↔ y(y x → (x Ay A)))
5 df-ral 2305 . . . . . . 7 (y x (x Ay A) ↔ y(y x → (x Ay A)))
64, 5bitr4i 176 . . . . . 6 (y((y x x A) → y A) ↔ y x (x Ay A))
7 r19.21v 2390 . . . . . 6 (y x (x Ay A) ↔ (x Ay x y A))
86, 7bitri 173 . . . . 5 (y((y x x A) → y A) ↔ (x Ay x y A))
98albii 1356 . . . 4 (xy((y x x A) → y A) ↔ x(x Ay x y A))
10 df-ral 2305 . . . 4 (x A y x y Ax(x Ay x y A))
119, 10bitr4i 176 . . 3 (xy((y x x A) → y A) ↔ x A y x y A)
122, 11bitri 173 . 2 (yx((y x x A) → y A) ↔ x A y x y A)
131, 12bitri 173 1 (Tr Ax A y x y A)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240   wcel 1390  wral 2300  Tr wtr 3845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-in 2918  df-ss 2925  df-uni 3572  df-tr 3846
This theorem is referenced by:  dftr3  3849
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