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Theorem oprabidlem 5479
Description: Slight elaboration of exdistrfor 1678. A lemma for oprabid 5480. (Contributed by Jim Kingdon, 15-Jan-2019.)
Assertion
Ref Expression
oprabidlem (xy(x = z ψ) → x(x = z yψ))
Distinct variable groups:   x,z   y,z
Allowed substitution hints:   ψ(x,y,z)

Proof of Theorem oprabidlem
StepHypRef Expression
1 ax-bnd 1396 . . 3 (y y = x (y y = z xy(x = zy x = z)))
2 ax-10 1393 . . . 4 (y y = xx x = y)
3 dtru 4238 . . . . . 6 ¬ y y = z
4 pm2.53 640 . . . . . 6 ((y y = z xy(x = zy x = z)) → (¬ y y = zxy(x = zy x = z)))
53, 4mpi 15 . . . . 5 ((y y = z xy(x = zy x = z)) → xy(x = zy x = z))
6 df-nf 1347 . . . . . 6 (Ⅎy x = zy(x = zy x = z))
76albii 1356 . . . . 5 (xy x = zxy(x = zy x = z))
85, 7sylibr 137 . . . 4 ((y y = z xy(x = zy x = z)) → xy x = z)
92, 8orim12i 675 . . 3 ((y y = x (y y = z xy(x = zy x = z))) → (x x = y xy x = z))
101, 9ax-mp 7 . 2 (x x = y xy x = z)
1110exdistrfor 1678 1 (xy(x = z ψ) → x(x = z yψ))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   wo 628  wal 1240  wnf 1346  wex 1378
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-in1 544  ax-in2 545  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-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-setind 4220
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-v 2553  df-dif 2914  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373
This theorem is referenced by:  oprabid  5480
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