Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  oprabidlem Structured version   GIF version

Theorem oprabidlem 5460
 Description: Slight elaboration of exdistrfor 1663. A lemma for oprabid 5461. (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 1380 . . 3 (y y = x (y y = z xy(x = zy x = z)))
2 ax-10 1377 . . . 4 (y y = xx x = y)
3 dtru 4222 . . . . . 6 ¬ y y = z
4 pm2.53 628 . . . . . 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 1330 . . . . . 6 (Ⅎy x = zy(x = zy x = z))
76albii 1339 . . . . 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 663 . . 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 1663 1 (xy(x = z ψ) → x(x = z yψ))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ∨ wo 616  ∀wal 1226  Ⅎwnf 1329  ∃wex 1362 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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-setind 4204 This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-v 2537  df-dif 2897  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356 This theorem is referenced by:  oprabid  5461
 Copyright terms: Public domain W3C validator