 Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax13lem1 Structured version   Visualization version   GIF version

Theorem ax13lem1 2247
 Description: A version of ax13v 2246 with one distinct variable restriction dropped. For convenience, 𝑦 is kept on the right side of equations. The proof of ax13 2248 bases on ideas from NM, 24-Dec-2015. (Contributed by Wolf Lammen, 8-Sep-2018.) (New usage is discouraged.)
Assertion
Ref Expression
ax13lem1 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
Distinct variable group:   𝑥,𝑧

Proof of Theorem ax13lem1
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 equviniva 1957 . 2 (𝑧 = 𝑦 → ∃𝑤(𝑧 = 𝑤𝑦 = 𝑤))
2 ax13v 2246 . . . . 5 𝑥 = 𝑦 → (𝑦 = 𝑤 → ∀𝑥 𝑦 = 𝑤))
3 equeucl 1948 . . . . . 6 (𝑧 = 𝑤 → (𝑦 = 𝑤𝑧 = 𝑦))
43alimdv 1842 . . . . 5 (𝑧 = 𝑤 → (∀𝑥 𝑦 = 𝑤 → ∀𝑥 𝑧 = 𝑦))
52, 4syl9 77 . . . 4 𝑥 = 𝑦 → (𝑧 = 𝑤 → (𝑦 = 𝑤 → ∀𝑥 𝑧 = 𝑦)))
65impd 447 . . 3 𝑥 = 𝑦 → ((𝑧 = 𝑤𝑦 = 𝑤) → ∀𝑥 𝑧 = 𝑦))
76exlimdv 1858 . 2 𝑥 = 𝑦 → (∃𝑤(𝑧 = 𝑤𝑦 = 𝑤) → ∀𝑥 𝑧 = 𝑦))
81, 7syl5 34 1 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384  ∀wal 1478  ∃wex 1701 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-13 2245 This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702 This theorem is referenced by:  ax13  2248  ax6e  2249  ax13lem2  2295  nfeqf2  2296  wl-19.8eqv  32941  wl-19.2reqv  32942  wl-dveeq12  32943
 Copyright terms: Public domain W3C validator