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

Theorem equveli 1686
Description: A variable elimination law for equality with no distinct variable requirements. (Compare equvini 1685.) (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 1419 . 2 (∀𝑧(𝑧 = 𝑥𝑧 = 𝑦) ↔ (∀𝑧(𝑧 = 𝑥𝑧 = 𝑦) ∧ ∀𝑧(𝑧 = 𝑦𝑧 = 𝑥)))
2 ax12or 1446 . . 3 (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
3 equequ1 1642 . . . . . . . . 9 (𝑧 = 𝑥 → (𝑧 = 𝑥𝑥 = 𝑥))
4 equequ1 1642 . . . . . . . . 9 (𝑧 = 𝑥 → (𝑧 = 𝑦𝑥 = 𝑦))
53, 4imbi12d 232 . . . . . . . 8 (𝑧 = 𝑥 → ((𝑧 = 𝑥𝑧 = 𝑦) ↔ (𝑥 = 𝑥𝑥 = 𝑦)))
65sps 1473 . . . . . . 7 (∀𝑧 𝑧 = 𝑥 → ((𝑧 = 𝑥𝑧 = 𝑦) ↔ (𝑥 = 𝑥𝑥 = 𝑦)))
76dral2 1663 . . . . . 6 (∀𝑧 𝑧 = 𝑥 → (∀𝑧(𝑧 = 𝑥𝑧 = 𝑦) ↔ ∀𝑧(𝑥 = 𝑥𝑥 = 𝑦)))
8 equid 1632 . . . . . . . . 9 𝑥 = 𝑥
98a1bi 241 . . . . . . . 8 (𝑥 = 𝑦 ↔ (𝑥 = 𝑥𝑥 = 𝑦))
109biimpri 131 . . . . . . 7 ((𝑥 = 𝑥𝑥 = 𝑦) → 𝑥 = 𝑦)
1110sps 1473 . . . . . 6 (∀𝑧(𝑥 = 𝑥𝑥 = 𝑦) → 𝑥 = 𝑦)
127, 11syl6bi 161 . . . . 5 (∀𝑧 𝑧 = 𝑥 → (∀𝑧(𝑧 = 𝑥𝑧 = 𝑦) → 𝑥 = 𝑦))
1312adantrd 273 . . . 4 (∀𝑧 𝑧 = 𝑥 → ((∀𝑧(𝑧 = 𝑥𝑧 = 𝑦) ∧ ∀𝑧(𝑧 = 𝑦𝑧 = 𝑥)) → 𝑥 = 𝑦))
14 equequ1 1642 . . . . . . . . . 10 (𝑧 = 𝑦 → (𝑧 = 𝑦𝑦 = 𝑦))
15 equequ1 1642 . . . . . . . . . 10 (𝑧 = 𝑦 → (𝑧 = 𝑥𝑦 = 𝑥))
1614, 15imbi12d 232 . . . . . . . . 9 (𝑧 = 𝑦 → ((𝑧 = 𝑦𝑧 = 𝑥) ↔ (𝑦 = 𝑦𝑦 = 𝑥)))
1716sps 1473 . . . . . . . 8 (∀𝑧 𝑧 = 𝑦 → ((𝑧 = 𝑦𝑧 = 𝑥) ↔ (𝑦 = 𝑦𝑦 = 𝑥)))
1817dral1 1662 . . . . . . 7 (∀𝑧 𝑧 = 𝑦 → (∀𝑧(𝑧 = 𝑦𝑧 = 𝑥) ↔ ∀𝑦(𝑦 = 𝑦𝑦 = 𝑥)))
19 equid 1632 . . . . . . . . 9 𝑦 = 𝑦
20 ax-4 1443 . . . . . . . . 9 (∀𝑦(𝑦 = 𝑦𝑦 = 𝑥) → (𝑦 = 𝑦𝑦 = 𝑥))
2119, 20mpi 15 . . . . . . . 8 (∀𝑦(𝑦 = 𝑦𝑦 = 𝑥) → 𝑦 = 𝑥)
22 equcomi 1635 . . . . . . . 8 (𝑦 = 𝑥𝑥 = 𝑦)
2321, 22syl 14 . . . . . . 7 (∀𝑦(𝑦 = 𝑦𝑦 = 𝑥) → 𝑥 = 𝑦)
2418, 23syl6bi 161 . . . . . 6 (∀𝑧 𝑧 = 𝑦 → (∀𝑧(𝑧 = 𝑦𝑧 = 𝑥) → 𝑥 = 𝑦))
2524adantld 272 . . . . 5 (∀𝑧 𝑧 = 𝑦 → ((∀𝑧(𝑧 = 𝑥𝑧 = 𝑦) ∧ ∀𝑧(𝑧 = 𝑦𝑧 = 𝑥)) → 𝑥 = 𝑦))
26 hba1 1476 . . . . . . . . . 10 (∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → ∀𝑧𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
27 hbequid 1449 . . . . . . . . . . 11 (𝑥 = 𝑥 → ∀𝑧 𝑥 = 𝑥)
2827a1i 9 . . . . . . . . . 10 (∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → (𝑥 = 𝑥 → ∀𝑧 𝑥 = 𝑥))
29 ax-4 1443 . . . . . . . . . 10 (∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
3026, 28, 29hbimd 1508 . . . . . . . . 9 (∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → ((𝑥 = 𝑥𝑥 = 𝑦) → ∀𝑧(𝑥 = 𝑥𝑥 = 𝑦)))
3130a5i 1478 . . . . . . . 8 (∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → ∀𝑧((𝑥 = 𝑥𝑥 = 𝑦) → ∀𝑧(𝑥 = 𝑥𝑥 = 𝑦)))
32 equtr 1639 . . . . . . . . . 10 (𝑧 = 𝑥 → (𝑥 = 𝑥𝑧 = 𝑥))
33 ax-8 1438 . . . . . . . . . 10 (𝑧 = 𝑥 → (𝑧 = 𝑦𝑥 = 𝑦))
3432, 33imim12d 73 . . . . . . . . 9 (𝑧 = 𝑥 → ((𝑧 = 𝑥𝑧 = 𝑦) → (𝑥 = 𝑥𝑥 = 𝑦)))
3534ax-gen 1381 . . . . . . . 8 𝑧(𝑧 = 𝑥 → ((𝑧 = 𝑥𝑧 = 𝑦) → (𝑥 = 𝑥𝑥 = 𝑦)))
36 19.26 1413 . . . . . . . . 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 1667 . . . . . . . . 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 133 . . . . . . . 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 404 . . . . . . 7 (∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → (∀𝑧(𝑧 = 𝑥𝑧 = 𝑦) → (𝑥 = 𝑥𝑥 = 𝑦)))
408, 39mpii 43 . . . . . 6 (∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → (∀𝑧(𝑧 = 𝑥𝑧 = 𝑦) → 𝑥 = 𝑦))
4140adantrd 273 . . . . 5 (∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → ((∀𝑧(𝑧 = 𝑥𝑧 = 𝑦) ∧ ∀𝑧(𝑧 = 𝑦𝑧 = 𝑥)) → 𝑥 = 𝑦))
4225, 41jaoi 669 . . . 4 ((∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)) → ((∀𝑧(𝑧 = 𝑥𝑧 = 𝑦) ∧ ∀𝑧(𝑧 = 𝑦𝑧 = 𝑥)) → 𝑥 = 𝑦))
4313, 42jaoi 669 . . 3 ((∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 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 119 1 (∀𝑧(𝑧 = 𝑥𝑧 = 𝑦) → 𝑥 = 𝑦)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wo 662  wal 1285   = wceq 1287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471
This theorem depends on definitions:  df-bi 115
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator