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Theorem equveli 1732
 Description: A variable elimination law for equality with no distinct variable requirements. (Compare equvini 1731.) (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 1463 . 2 (∀𝑧(𝑧 = 𝑥𝑧 = 𝑦) ↔ (∀𝑧(𝑧 = 𝑥𝑧 = 𝑦) ∧ ∀𝑧(𝑧 = 𝑦𝑧 = 𝑥)))
2 ax12or 1490 . . 3 (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
3 equequ1 1688 . . . . . . . . 9 (𝑧 = 𝑥 → (𝑧 = 𝑥𝑥 = 𝑥))
4 equequ1 1688 . . . . . . . . 9 (𝑧 = 𝑥 → (𝑧 = 𝑦𝑥 = 𝑦))
53, 4imbi12d 233 . . . . . . . 8 (𝑧 = 𝑥 → ((𝑧 = 𝑥𝑧 = 𝑦) ↔ (𝑥 = 𝑥𝑥 = 𝑦)))
65sps 1517 . . . . . . 7 (∀𝑧 𝑧 = 𝑥 → ((𝑧 = 𝑥𝑧 = 𝑦) ↔ (𝑥 = 𝑥𝑥 = 𝑦)))
76dral2 1709 . . . . . 6 (∀𝑧 𝑧 = 𝑥 → (∀𝑧(𝑧 = 𝑥𝑧 = 𝑦) ↔ ∀𝑧(𝑥 = 𝑥𝑥 = 𝑦)))
8 equid 1677 . . . . . . . . 9 𝑥 = 𝑥
98a1bi 242 . . . . . . . 8 (𝑥 = 𝑦 ↔ (𝑥 = 𝑥𝑥 = 𝑦))
109biimpri 132 . . . . . . 7 ((𝑥 = 𝑥𝑥 = 𝑦) → 𝑥 = 𝑦)
1110sps 1517 . . . . . 6 (∀𝑧(𝑥 = 𝑥𝑥 = 𝑦) → 𝑥 = 𝑦)
127, 11syl6bi 162 . . . . 5 (∀𝑧 𝑧 = 𝑥 → (∀𝑧(𝑧 = 𝑥𝑧 = 𝑦) → 𝑥 = 𝑦))
1312adantrd 277 . . . 4 (∀𝑧 𝑧 = 𝑥 → ((∀𝑧(𝑧 = 𝑥𝑧 = 𝑦) ∧ ∀𝑧(𝑧 = 𝑦𝑧 = 𝑥)) → 𝑥 = 𝑦))
14 equequ1 1688 . . . . . . . . . 10 (𝑧 = 𝑦 → (𝑧 = 𝑦𝑦 = 𝑦))
15 equequ1 1688 . . . . . . . . . 10 (𝑧 = 𝑦 → (𝑧 = 𝑥𝑦 = 𝑥))
1614, 15imbi12d 233 . . . . . . . . 9 (𝑧 = 𝑦 → ((𝑧 = 𝑦𝑧 = 𝑥) ↔ (𝑦 = 𝑦𝑦 = 𝑥)))
1716sps 1517 . . . . . . . 8 (∀𝑧 𝑧 = 𝑦 → ((𝑧 = 𝑦𝑧 = 𝑥) ↔ (𝑦 = 𝑦𝑦 = 𝑥)))
1817dral1 1708 . . . . . . 7 (∀𝑧 𝑧 = 𝑦 → (∀𝑧(𝑧 = 𝑦𝑧 = 𝑥) ↔ ∀𝑦(𝑦 = 𝑦𝑦 = 𝑥)))
19 equid 1677 . . . . . . . . 9 𝑦 = 𝑦
20 ax-4 1487 . . . . . . . . 9 (∀𝑦(𝑦 = 𝑦𝑦 = 𝑥) → (𝑦 = 𝑦𝑦 = 𝑥))
2119, 20mpi 15 . . . . . . . 8 (∀𝑦(𝑦 = 𝑦𝑦 = 𝑥) → 𝑦 = 𝑥)
22 equcomi 1680 . . . . . . . 8 (𝑦 = 𝑥𝑥 = 𝑦)
2321, 22syl 14 . . . . . . 7 (∀𝑦(𝑦 = 𝑦𝑦 = 𝑥) → 𝑥 = 𝑦)
2418, 23syl6bi 162 . . . . . 6 (∀𝑧 𝑧 = 𝑦 → (∀𝑧(𝑧 = 𝑦𝑧 = 𝑥) → 𝑥 = 𝑦))
2524adantld 276 . . . . 5 (∀𝑧 𝑧 = 𝑦 → ((∀𝑧(𝑧 = 𝑥𝑧 = 𝑦) ∧ ∀𝑧(𝑧 = 𝑦𝑧 = 𝑥)) → 𝑥 = 𝑦))
26 hba1 1520 . . . . . . . . . 10 (∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → ∀𝑧𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
27 hbequid 1493 . . . . . . . . . . 11 (𝑥 = 𝑥 → ∀𝑧 𝑥 = 𝑥)
2827a1i 9 . . . . . . . . . 10 (∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → (𝑥 = 𝑥 → ∀𝑧 𝑥 = 𝑥))
29 ax-4 1487 . . . . . . . . . 10 (∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
3026, 28, 29hbimd 1552 . . . . . . . . 9 (∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → ((𝑥 = 𝑥𝑥 = 𝑦) → ∀𝑧(𝑥 = 𝑥𝑥 = 𝑦)))
3130a5i 1522 . . . . . . . 8 (∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → ∀𝑧((𝑥 = 𝑥𝑥 = 𝑦) → ∀𝑧(𝑥 = 𝑥𝑥 = 𝑦)))
32 equtr 1685 . . . . . . . . . 10 (𝑧 = 𝑥 → (𝑥 = 𝑥𝑧 = 𝑥))
33 ax-8 1482 . . . . . . . . . 10 (𝑧 = 𝑥 → (𝑧 = 𝑦𝑥 = 𝑦))
3432, 33imim12d 74 . . . . . . . . 9 (𝑧 = 𝑥 → ((𝑧 = 𝑥𝑧 = 𝑦) → (𝑥 = 𝑥𝑥 = 𝑦)))
3534ax-gen 1425 . . . . . . . 8 𝑧(𝑧 = 𝑥 → ((𝑧 = 𝑥𝑧 = 𝑦) → (𝑥 = 𝑥𝑥 = 𝑦)))
36 19.26 1457 . . . . . . . . 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 1713 . . . . . . . . 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 134 . . . . . . . 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 409 . . . . . . 7 (∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → (∀𝑧(𝑧 = 𝑥𝑧 = 𝑦) → (𝑥 = 𝑥𝑥 = 𝑦)))
408, 39mpii 44 . . . . . 6 (∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → (∀𝑧(𝑧 = 𝑥𝑧 = 𝑦) → 𝑥 = 𝑦))
4140adantrd 277 . . . . 5 (∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → ((∀𝑧(𝑧 = 𝑥𝑧 = 𝑦) ∧ ∀𝑧(𝑧 = 𝑦𝑧 = 𝑥)) → 𝑥 = 𝑦))
4225, 41jaoi 705 . . . 4 ((∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)) → ((∀𝑧(𝑧 = 𝑥𝑧 = 𝑦) ∧ ∀𝑧(𝑧 = 𝑦𝑧 = 𝑥)) → 𝑥 = 𝑦))
4313, 42jaoi 705 . . 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 5 . 2 ((∀𝑧(𝑧 = 𝑥𝑧 = 𝑦) ∧ ∀𝑧(𝑧 = 𝑦𝑧 = 𝑥)) → 𝑥 = 𝑦)
451, 44sylbi 120 1 (∀𝑧(𝑧 = 𝑥𝑧 = 𝑦) → 𝑥 = 𝑦)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 697  ∀wal 1329   = wceq 1331 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515 This theorem depends on definitions:  df-bi 116 This theorem is referenced by: (None)
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