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Theorem equveli 1624
Description: A variable elimination law for equality with no distinct variable requirements. (Compare equvini 1623.) (Contributed by NM, 1-Mar-2013.) (Revised by NM, 3-Feb-2015.)
Assertion
Ref Expression
equveli (z(z = xz = y) → x = y)

Proof of Theorem equveli
StepHypRef Expression
1 albiim 1357 . 2 (z(z = xz = y) ↔ (z(z = xz = y) z(z = yz = x)))
2 ax12or 1384 . . 3 (z z = x (z z = y z(x = yz x = y)))
3 equequ1 1580 . . . . . . . . 9 (z = x → (z = xx = x))
4 equequ1 1580 . . . . . . . . 9 (z = x → (z = yx = y))
53, 4imbi12d 223 . . . . . . . 8 (z = x → ((z = xz = y) ↔ (x = xx = y)))
65sps 1412 . . . . . . 7 (z z = x → ((z = xz = y) ↔ (x = xx = y)))
76dral2 1601 . . . . . 6 (z z = x → (z(z = xz = y) ↔ z(x = xx = y)))
8 equid 1571 . . . . . . . . 9 x = x
98a1bi 232 . . . . . . . 8 (x = y ↔ (x = xx = y))
109biimpri 124 . . . . . . 7 ((x = xx = y) → x = y)
1110sps 1412 . . . . . 6 (z(x = xx = y) → x = y)
127, 11syl6bi 152 . . . . 5 (z z = x → (z(z = xz = y) → x = y))
1312adantrd 264 . . . 4 (z z = x → ((z(z = xz = y) z(z = yz = x)) → x = y))
14 equequ1 1580 . . . . . . . . . 10 (z = y → (z = yy = y))
15 equequ1 1580 . . . . . . . . . 10 (z = y → (z = xy = x))
1614, 15imbi12d 223 . . . . . . . . 9 (z = y → ((z = yz = x) ↔ (y = yy = x)))
1716sps 1412 . . . . . . . 8 (z z = y → ((z = yz = x) ↔ (y = yy = x)))
1817dral1 1600 . . . . . . 7 (z z = y → (z(z = yz = x) ↔ y(y = yy = x)))
19 equid 1571 . . . . . . . . 9 y = y
20 ax-4 1381 . . . . . . . . 9 (y(y = yy = x) → (y = yy = x))
2119, 20mpi 15 . . . . . . . 8 (y(y = yy = x) → y = x)
22 equcomi 1574 . . . . . . . 8 (y = xx = y)
2321, 22syl 14 . . . . . . 7 (y(y = yy = x) → x = y)
2418, 23syl6bi 152 . . . . . 6 (z z = y → (z(z = yz = x) → x = y))
2524adantld 263 . . . . 5 (z z = y → ((z(z = xz = y) z(z = yz = x)) → x = y))
26 hba1 1415 . . . . . . . . . 10 (z(x = yz x = y) → zz(x = yz x = y))
27 hbequid 1387 . . . . . . . . . . 11 (x = xz x = x)
2827a1i 9 . . . . . . . . . 10 (z(x = yz x = y) → (x = xz x = x))
29 ax-4 1381 . . . . . . . . . 10 (z(x = yz x = y) → (x = yz x = y))
3026, 28, 29hbimd 1447 . . . . . . . . 9 (z(x = yz x = y) → ((x = xx = y) → z(x = xx = y)))
3130a5i 1417 . . . . . . . 8 (z(x = yz x = y) → z((x = xx = y) → z(x = xx = y)))
32 equtr 1577 . . . . . . . . . 10 (z = x → (x = xz = x))
33 ax-8 1376 . . . . . . . . . 10 (z = x → (z = yx = y))
3432, 33imim12d 68 . . . . . . . . 9 (z = x → ((z = xz = y) → (x = xx = y)))
3534ax-gen 1318 . . . . . . . 8 z(z = x → ((z = xz = y) → (x = xx = y)))
36 19.26 1350 . . . . . . . . 9 (z(((x = xx = y) → z(x = xx = y)) (z = x → ((z = xz = y) → (x = xx = y)))) ↔ (z((x = xx = y) → z(x = xx = y)) z(z = x → ((z = xz = y) → (x = xx = y)))))
37 spimth 1605 . . . . . . . . 9 (z(((x = xx = y) → z(x = xx = y)) (z = x → ((z = xz = y) → (x = xx = y)))) → (z(z = xz = y) → (x = xx = y)))
3836, 37sylbir 125 . . . . . . . 8 ((z((x = xx = y) → z(x = xx = y)) z(z = x → ((z = xz = y) → (x = xx = y)))) → (z(z = xz = y) → (x = xx = y)))
3931, 35, 38sylancl 394 . . . . . . 7 (z(x = yz x = y) → (z(z = xz = y) → (x = xx = y)))
408, 39mpii 39 . . . . . 6 (z(x = yz x = y) → (z(z = xz = y) → x = y))
4140adantrd 264 . . . . 5 (z(x = yz x = y) → ((z(z = xz = y) z(z = yz = x)) → x = y))
4225, 41jaoi 623 . . . 4 ((z z = y z(x = yz x = y)) → ((z(z = xz = y) z(z = yz = x)) → x = y))
4313, 42jaoi 623 . . 3 ((z z = x (z z = y z(x = yz x = y))) → ((z(z = xz = y) z(z = yz = x)) → x = y))
442, 43ax-mp 7 . 2 ((z(z = xz = y) z(z = yz = x)) → x = y)
451, 44sylbi 114 1 (z(z = xz = y) → x = y)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   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-ie1 1363  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  ax-i5r 1410
This theorem depends on definitions:  df-bi 110
This theorem is referenced by: (None)
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