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

Theorem equvini 2467
 Description: A variable introduction law for equality. Lemma 15 of [Monk2] p. 109, however we do not require 𝑧 to be distinct from 𝑥 and 𝑦. Usage of this theorem is discouraged because it depends on ax-13 2380. See equvinv 2037 for a shorter proof requiring fewer axioms when 𝑧 is required to be distinct from 𝑥 and 𝑦. (Contributed by NM, 10-Jan-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 16-Sep-2023.) (New usage is discouraged.)
Assertion
Ref Expression
equvini (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧𝑧 = 𝑦))

Proof of Theorem equvini
StepHypRef Expression
1 equtr 2029 . . . 4 (𝑧 = 𝑥 → (𝑥 = 𝑦𝑧 = 𝑦))
2 equcomi 2025 . . . 4 (𝑧 = 𝑥𝑥 = 𝑧)
31, 2jctild 530 . . 3 (𝑧 = 𝑥 → (𝑥 = 𝑦 → (𝑥 = 𝑧𝑧 = 𝑦)))
4 19.8a 2179 . . 3 ((𝑥 = 𝑧𝑧 = 𝑦) → ∃𝑧(𝑥 = 𝑧𝑧 = 𝑦))
53, 4syl6 35 . 2 (𝑧 = 𝑥 → (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧𝑧 = 𝑦)))
6 ax13 2383 . . 3 𝑧 = 𝑥 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
7 ax6e 2391 . . . . 5 𝑧 𝑧 = 𝑥
87, 3eximii 1839 . . . 4 𝑧(𝑥 = 𝑦 → (𝑥 = 𝑧𝑧 = 𝑦))
9819.35i 1880 . . 3 (∀𝑧 𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧𝑧 = 𝑦))
106, 9syl6 35 . 2 𝑧 = 𝑥 → (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧𝑧 = 𝑦)))
115, 10pm2.61i 185 1 (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧𝑧 = 𝑦))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400  ∀wal 1537  ∃wex 1782 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-12 2176  ax-13 2380 This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1783 This theorem is referenced by:  2ax6elem  2483
 Copyright terms: Public domain W3C validator