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Theorem equvini 1619
Description: A variable introduction law for equality. Lemma 15 of [Monk2] p. 109, however we do not require z to be distinct from x and y (making the proof longer). (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Assertion
Ref Expression
equvini (x = yz(x = z z = y))

Proof of Theorem equvini
StepHypRef Expression
1 ax12or 1380 . 2 (z z = x (z z = y z(x = yz x = y)))
2 equcomi 1570 . . . . . . 7 (z = xx = z)
32alimi 1320 . . . . . 6 (z z = xz x = z)
4 a9e 1564 . . . . . 6 z z = y
53, 4jctir 296 . . . . 5 (z z = x → (z x = z z z = y))
65a1d 22 . . . 4 (z z = x → (x = y → (z x = z z z = y)))
7 19.29 1489 . . . 4 ((z x = z z z = y) → z(x = z z = y))
86, 7syl6 29 . . 3 (z z = x → (x = yz(x = z z = y)))
9 a9e 1564 . . . . . . . 8 z z = x
102eximi 1469 . . . . . . . 8 (z z = xz x = z)
119, 10ax-mp 7 . . . . . . 7 z x = z
1211a1ii 24 . . . . . 6 (z z = y → (x = yz x = z))
1312anc2ri 313 . . . . 5 (z z = y → (x = y → (z x = z z z = y)))
14 19.29r 1490 . . . . 5 ((z x = z z z = y) → z(x = z z = y))
1513, 14syl6 29 . . . 4 (z z = y → (x = yz(x = z z = y)))
16 ax-8 1372 . . . . . . . . . . . 12 (x = z → (x = yz = y))
1716anc2li 312 . . . . . . . . . . 11 (x = z → (x = y → (x = z z = y)))
1817equcoms 1572 . . . . . . . . . 10 (z = x → (x = y → (x = z z = y)))
1918com12 27 . . . . . . . . 9 (x = y → (z = x → (x = z z = y)))
2019alimi 1320 . . . . . . . 8 (z x = yz(z = x → (x = z z = y)))
21 exim 1468 . . . . . . . 8 (z(z = x → (x = z z = y)) → (z z = xz(x = z z = y)))
2220, 21syl 14 . . . . . . 7 (z x = y → (z z = xz(x = z z = y)))
239, 22mpi 15 . . . . . 6 (z x = yz(x = z z = y))
2423imim2i 12 . . . . 5 ((x = yz x = y) → (x = yz(x = z z = y)))
2524sps 1408 . . . 4 (z(x = yz x = y) → (x = yz(x = z z = y)))
2615, 25jaoi 623 . . 3 ((z z = y z(x = yz x = y)) → (x = yz(x = z z = y)))
278, 26jaoi 623 . 2 ((z z = x (z z = y z(x = yz x = y))) → (x = yz(x = z z = y)))
281, 27ax-mp 7 1 (x = yz(x = z z = y))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wo 616  wal 1224   = wceq 1226  wex 1358
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 1312  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-i12 1375  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405
This theorem depends on definitions:  df-bi 110
This theorem is referenced by:  sbequi  1698  equvin  1721
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