Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  equvel Structured version   Visualization version   GIF version

Theorem equvel 2335
 Description: A variable elimination law for equality with no distinct variable requirements. Compare equvini 2334. (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 1736 . 2 (∀𝑧(𝑧 = 𝑥𝑧 = 𝑦) → (∀𝑧 𝑧 = 𝑥 ↔ ∀𝑧 𝑧 = 𝑦))
2 ax6e 2238 . . . 4 𝑧 𝑧 = 𝑦
3 biimpr 209 . . . . . 6 ((𝑧 = 𝑥𝑧 = 𝑦) → (𝑧 = 𝑦𝑧 = 𝑥))
4 ax7 1930 . . . . . 6 (𝑧 = 𝑥 → (𝑧 = 𝑦𝑥 = 𝑦))
53, 4syli 38 . . . . 5 ((𝑧 = 𝑥𝑧 = 𝑦) → (𝑧 = 𝑦𝑥 = 𝑦))
65com12 32 . . . 4 (𝑧 = 𝑦 → ((𝑧 = 𝑥𝑧 = 𝑦) → 𝑥 = 𝑦))
72, 6eximii 1754 . . 3 𝑧((𝑧 = 𝑥𝑧 = 𝑦) → 𝑥 = 𝑦)
8719.35i 1795 . 2 (∀𝑧(𝑧 = 𝑥𝑧 = 𝑦) → ∃𝑧 𝑥 = 𝑦)
94spsd 2045 . . . . 5 (𝑧 = 𝑥 → (∀𝑧 𝑧 = 𝑦𝑥 = 𝑦))
109sps 2043 . . . 4 (∀𝑧 𝑧 = 𝑥 → (∀𝑧 𝑧 = 𝑦𝑥 = 𝑦))
1110a1dd 48 . . 3 (∀𝑧 𝑧 = 𝑥 → (∀𝑧 𝑧 = 𝑦 → (∃𝑧 𝑥 = 𝑦𝑥 = 𝑦)))
12 nfeqf 2289 . . . . 5 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧 𝑥 = 𝑦)
131219.9d 2058 . . . 4 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (∃𝑧 𝑥 = 𝑦𝑥 = 𝑦))
1413ex 449 . . 3 (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (∃𝑧 𝑥 = 𝑦𝑥 = 𝑦)))
1511, 14bija 369 . 2 ((∀𝑧 𝑧 = 𝑥 ↔ ∀𝑧 𝑧 = 𝑦) → (∃𝑧 𝑥 = 𝑦𝑥 = 𝑦))
161, 8, 15sylc 63 1 (∀𝑧(𝑧 = 𝑥𝑧 = 𝑦) → 𝑥 = 𝑦)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473  ∃wex 1695 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034  ax-13 2234 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator