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Theorem axc11nOLDOLD 2297
 Description: Old proof of axc11n 2295. Obsolete as of 29-Mar-2021. (Contributed by NM, 10-May-1993.) (Revised by NM, 7-Nov-2015.) (Proof shortened by Wolf Lammen, 6-Mar-2018.) Adapt to a modification of axc11nlemOLD2 1975. (Revised by Wolf Lammen, 30-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
axc11nOLDOLD (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Proof of Theorem axc11nOLDOLD
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 equcomi 1931 . . . . 5 (𝑧 = 𝑥𝑥 = 𝑧)
2 dveeq1 2288 . . . . 5 (¬ ∀𝑦 𝑦 = 𝑥 → (𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧))
31, 2syl5com 31 . . . 4 (𝑧 = 𝑥 → (¬ ∀𝑦 𝑦 = 𝑥 → ∀𝑦 𝑥 = 𝑧))
4 axc11r 2175 . . . . 5 (∀𝑥 𝑥 = 𝑦 → (∀𝑦 𝑥 = 𝑧 → ∀𝑥 𝑥 = 𝑧))
52axc11nlemOLD2 1975 . . . . 5 (∀𝑥 𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑥)
64, 5syl6 34 . . . 4 (∀𝑥 𝑥 = 𝑦 → (∀𝑦 𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑥))
73, 6syl9 75 . . 3 (𝑧 = 𝑥 → (∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑦 𝑦 = 𝑥 → ∀𝑦 𝑦 = 𝑥)))
8 ax6ev 1877 . . 3 𝑧 𝑧 = 𝑥
97, 8exlimiiv 1846 . 2 (∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑦 𝑦 = 𝑥 → ∀𝑦 𝑦 = 𝑥))
109pm2.18d 123 1 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1473 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)
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