 Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  axc11nlemOLD2 Structured version   Visualization version   GIF version

Theorem axc11nlemOLD2 1990
 Description: Lemma for axc11n 2311. Change bound variable in an equality. Obsolete as of 29-Mar-2021. Use aev 1985 instead. (Contributed by NM, 8-Jul-2016.) (Proof shortened by Wolf Lammen, 17-Feb-2018.) Restructure to ease either bundling, or reducing dependencies on axioms. (Revised by Wolf Lammen, 30-Nov-2019.) Remove dependency on ax-12 2049. (Revised by Wolf Lammen, 14-Mar-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
axc11nlemOLD2.1 (¬ ∀𝑦 𝑦 = 𝑥 → (𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧))
Assertion
Ref Expression
axc11nlemOLD2 (∀𝑥 𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑥)
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem axc11nlemOLD2
StepHypRef Expression
1 cbvaev 1981 . . 3 (∀𝑥 𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑧)
2 equequ2 1955 . . . . 5 (𝑥 = 𝑧 → (𝑦 = 𝑥𝑦 = 𝑧))
32biimprd 238 . . . 4 (𝑥 = 𝑧 → (𝑦 = 𝑧𝑦 = 𝑥))
43al2imi 1740 . . 3 (∀𝑦 𝑥 = 𝑧 → (∀𝑦 𝑦 = 𝑧 → ∀𝑦 𝑦 = 𝑥))
51, 4syl5com 31 . 2 (∀𝑥 𝑥 = 𝑧 → (∀𝑦 𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑥))
6 spaev 1980 . . . 4 (∀𝑥 𝑥 = 𝑧𝑥 = 𝑧)
7 axc11nlemOLD2.1 . . . 4 (¬ ∀𝑦 𝑦 = 𝑥 → (𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧))
86, 7syl5com 31 . . 3 (∀𝑥 𝑥 = 𝑧 → (¬ ∀𝑦 𝑦 = 𝑥 → ∀𝑦 𝑥 = 𝑧))
98con1d 139 . 2 (∀𝑥 𝑥 = 𝑧 → (¬ ∀𝑦 𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑥))
105, 9pm2.61d 170 1 (∀𝑥 𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑥)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1478 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 1841  ax-6 1890  ax-7 1937 This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702 This theorem is referenced by:  aevlemOLD  1991  axc11nOLDOLD  2313
 Copyright terms: Public domain W3C validator