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Theorem equveli 1988
Description: A variable elimination law for equality with no distinct variable requirements. (Compare equvini 1987.) (Contributed by NM, 1-Mar-2013.) (Proof shortened by Mario Carneiro, 17-Oct-2016.)
Assertion
Ref Expression
equveli (z(z = xz = y) → x = y)

Proof of Theorem equveli
StepHypRef Expression
1 albiim 1611 . 2 (z(z = xz = y) ↔ (z(z = xz = y) z(z = yz = x)))
2 equequ1 1684 . . . . . . . 8 (z = y → (z = yy = y))
3 equequ1 1684 . . . . . . . 8 (z = y → (z = xy = x))
42, 3imbi12d 311 . . . . . . 7 (z = y → ((z = yz = x) ↔ (y = yy = x)))
54sps 1754 . . . . . 6 (z z = y → ((z = yz = x) ↔ (y = yy = x)))
65dral1 1965 . . . . 5 (z z = y → (z(z = yz = x) ↔ y(y = yy = x)))
7 equid 1676 . . . . . . 7 y = y
8 sp 1747 . . . . . . 7 (y(y = yy = x) → (y = yy = x))
97, 8mpi 16 . . . . . 6 (y(y = yy = x) → y = x)
10 equcomi 1679 . . . . . 6 (y = xx = y)
119, 10syl 15 . . . . 5 (y(y = yy = x) → x = y)
126, 11syl6bi 219 . . . 4 (z z = y → (z(z = yz = x) → x = y))
1312adantld 453 . . 3 (z z = y → ((z(z = xz = y) z(z = yz = x)) → x = y))
14 equequ1 1684 . . . . . . . . . 10 (z = x → (z = xx = x))
15 equequ1 1684 . . . . . . . . . 10 (z = x → (z = yx = y))
1614, 15imbi12d 311 . . . . . . . . 9 (z = x → ((z = xz = y) ↔ (x = xx = y)))
1716sps 1754 . . . . . . . 8 (z z = x → ((z = xz = y) ↔ (x = xx = y)))
1817dral2 1966 . . . . . . 7 (z z = x → (z(z = xz = y) ↔ z(x = xx = y)))
19 equid 1676 . . . . . . . . . 10 x = x
2019a1bi 327 . . . . . . . . 9 (x = y ↔ (x = xx = y))
2120biimpri 197 . . . . . . . 8 ((x = xx = y) → x = y)
2221sps 1754 . . . . . . 7 (z(x = xx = y) → x = y)
2318, 22syl6bi 219 . . . . . 6 (z z = x → (z(z = xz = y) → x = y))
2423a1d 22 . . . . 5 (z z = x → (¬ z z = y → (z(z = xz = y) → x = y)))
25 nfeqf 1958 . . . . . . 7 ((¬ z z = x ¬ z z = y) → Ⅎz x = y)
26 equtr 1682 . . . . . . . . . 10 (z = x → (x = xz = x))
27 ax-8 1675 . . . . . . . . . 10 (z = x → (z = yx = y))
2826, 27imim12d 68 . . . . . . . . 9 (z = x → ((z = xz = y) → (x = xx = y)))
2919, 28mpii 39 . . . . . . . 8 (z = x → ((z = xz = y) → x = y))
3029ax-gen 1546 . . . . . . 7 z(z = x → ((z = xz = y) → x = y))
31 spimt 1974 . . . . . . 7 ((Ⅎz x = y z(z = x → ((z = xz = y) → x = y))) → (z(z = xz = y) → x = y))
3225, 30, 31sylancl 643 . . . . . 6 ((¬ z z = x ¬ z z = y) → (z(z = xz = y) → x = y))
3332ex 423 . . . . 5 z z = x → (¬ z z = y → (z(z = xz = y) → x = y)))
3424, 33pm2.61i 156 . . . 4 z z = y → (z(z = xz = y) → x = y))
3534adantrd 454 . . 3 z z = y → ((z(z = xz = y) z(z = yz = x)) → x = y))
3613, 35pm2.61i 156 . 2 ((z(z = xz = y) z(z = yz = x)) → x = y)
371, 36sylbi 187 1 (z(z = xz = y) → x = y)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 176   wa 358  wal 1540  wnf 1544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545
This theorem is referenced by: (None)
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