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Theorem axc11n 2421
Description: Derive set.mm's original ax-c11n 38360 from others. Commutation law for identical variable specifiers. The antecedent and consequent are true when 𝑥 and 𝑦 are substituted with the same variable. Lemma L12 in [Megill] p. 445 (p. 12 of the preprint). If a disjoint variable condition is added on 𝑥 and 𝑦, then this becomes an instance of aevlem 2051. Use aecom 2422 instead when this does not lengthen the proof. Usage of this theorem is discouraged because it depends on ax-13 2367. (Contributed by NM, 10-May-1993.) (Revised by NM, 7-Nov-2015.) (Proof shortened by Wolf Lammen, 6-Mar-2018.) (Revised by Wolf Lammen, 30-Nov-2019.) (Proof shortened by BJ, 29-Mar-2021.) (Proof shortened by Wolf Lammen, 2-Jul-2021.) (New usage is discouraged.)
Assertion
Ref Expression
axc11n (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Proof of Theorem axc11n
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 dveeq1 2375 . . . . 5 (¬ ∀𝑦 𝑦 = 𝑥 → (𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧))
21com12 32 . . . 4 (𝑥 = 𝑧 → (¬ ∀𝑦 𝑦 = 𝑥 → ∀𝑦 𝑥 = 𝑧))
3 axc11r 2361 . . . . 5 (∀𝑥 𝑥 = 𝑦 → (∀𝑦 𝑥 = 𝑧 → ∀𝑥 𝑥 = 𝑧))
4 aev 2053 . . . . 5 (∀𝑥 𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑥)
53, 4syl6 35 . . . 4 (∀𝑥 𝑥 = 𝑦 → (∀𝑦 𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑥))
62, 5syl9 77 . . 3 (𝑥 = 𝑧 → (∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑦 𝑦 = 𝑥 → ∀𝑦 𝑦 = 𝑥)))
7 ax6evr 2011 . . 3 𝑧 𝑥 = 𝑧
86, 7exlimiiv 1927 . 2 (∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑦 𝑦 = 𝑥 → ∀𝑦 𝑦 = 𝑥))
98pm2.18d 127 1 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-10 2130  ax-12 2167  ax-13 2367
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1775  df-nf 1779
This theorem is referenced by:  aecom  2422  axi10  2696  wl-hbae1  36986  wl-ax11-lem3  37054  wl-ax11-lem8  37059  2sb5ndVD  44349  e2ebindVD  44351  e2ebindALT  44368  2sb5ndALT  44371
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