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Theorem hbae 1588
Description: All variables are effectively bound in an identical variable specifier. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)
Assertion
Ref Expression
hbae (x x = yzx x = y)

Proof of Theorem hbae
StepHypRef Expression
1 ax12or 1384 . . . 4 (z z = x (z z = y z(x = yz x = y)))
2 ax10o 1585 . . . . . 6 (x x = z → (x x = yz x = y))
32alequcoms 1390 . . . . 5 (z z = x → (x x = yz x = y))
4 ax10o 1585 . . . . . . . . 9 (x x = y → (x x = yy x = y))
54pm2.43i 43 . . . . . . . 8 (x x = yy x = y)
6 ax10o 1585 . . . . . . . 8 (y y = z → (y x = yz x = y))
75, 6syl5 28 . . . . . . 7 (y y = z → (x x = yz x = y))
87alequcoms 1390 . . . . . 6 (z z = y → (x x = yz x = y))
9 ax-4 1381 . . . . . . . 8 (x x = yx = y)
109imim1i 54 . . . . . . 7 ((x = yz x = y) → (x x = yz x = y))
1110sps 1412 . . . . . 6 (z(x = yz x = y) → (x x = yz x = y))
128, 11jaoi 623 . . . . 5 ((z z = y z(x = yz x = y)) → (x x = yz x = y))
133, 12jaoi 623 . . . 4 ((z z = x (z z = y z(x = yz x = y))) → (x x = yz x = y))
141, 13ax-mp 7 . . 3 (x x = yz x = y)
1514a5i 1417 . 2 (x x = yxz x = y)
16 ax-7 1317 . 2 (xz x = yzx x = y)
1715, 16syl 14 1 (x x = yzx x = y)
Colors of variables: wff set class
Syntax hints:  wi 4   wo 616  wal 1226   = wceq 1228
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409
This theorem depends on definitions:  df-bi 110
This theorem is referenced by:  nfae  1589  hbaes  1590  hbnae  1591  dral1  1600  dral2  1601  drex2  1602  drex1  1661  aev  1675  sbcomxyyz  1828  exists1  1978
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