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Theorem equvel 2471
Description: A variable elimination law for equality with no distinct variable requirements. Compare equvini 2469. (Contributed by NM, 1-Mar-2013.) (Proof shortened by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 15-Jun-2019.)
Assertion
Ref Expression
equvel (∀𝑧(𝑧 = 𝑥𝑧 = 𝑦) → 𝑥 = 𝑦)

Proof of Theorem equvel
StepHypRef Expression
1 albi 1810 . 2 (∀𝑧(𝑧 = 𝑥𝑧 = 𝑦) → (∀𝑧 𝑧 = 𝑥 ↔ ∀𝑧 𝑧 = 𝑦))
2 ax6e 2392 . . . 4 𝑧 𝑧 = 𝑦
3 biimpr 221 . . . . . 6 ((𝑧 = 𝑥𝑧 = 𝑦) → (𝑧 = 𝑦𝑧 = 𝑥))
4 ax7 2014 . . . . . 6 (𝑧 = 𝑥 → (𝑧 = 𝑦𝑥 = 𝑦))
53, 4syli 39 . . . . 5 ((𝑧 = 𝑥𝑧 = 𝑦) → (𝑧 = 𝑦𝑥 = 𝑦))
65com12 32 . . . 4 (𝑧 = 𝑦 → ((𝑧 = 𝑥𝑧 = 𝑦) → 𝑥 = 𝑦))
72, 6eximii 1828 . . 3 𝑧((𝑧 = 𝑥𝑧 = 𝑦) → 𝑥 = 𝑦)
8719.35i 1870 . 2 (∀𝑧(𝑧 = 𝑥𝑧 = 𝑦) → ∃𝑧 𝑥 = 𝑦)
94spsd 2176 . . . . 5 (𝑧 = 𝑥 → (∀𝑧 𝑧 = 𝑦𝑥 = 𝑦))
109sps 2174 . . . 4 (∀𝑧 𝑧 = 𝑥 → (∀𝑧 𝑧 = 𝑦𝑥 = 𝑦))
1110a1dd 50 . . 3 (∀𝑧 𝑧 = 𝑥 → (∀𝑧 𝑧 = 𝑦 → (∃𝑧 𝑥 = 𝑦𝑥 = 𝑦)))
12 nfeqf 2390 . . . . 5 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧 𝑥 = 𝑦)
131219.9d 2193 . . . 4 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (∃𝑧 𝑥 = 𝑦𝑥 = 𝑦))
1413ex 413 . . 3 (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (∃𝑧 𝑥 = 𝑦𝑥 = 𝑦)))
1511, 14bija 382 . 2 ((∀𝑧 𝑧 = 𝑥 ↔ ∀𝑧 𝑧 = 𝑦) → (∃𝑧 𝑥 = 𝑦𝑥 = 𝑦))
161, 8, 15sylc 65 1 (∀𝑧(𝑧 = 𝑥𝑧 = 𝑦) → 𝑥 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wal 1526  wex 1771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-10 2136  ax-12 2167  ax-13 2381
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776
This theorem is referenced by: (None)
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