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Theorem axc11n 2462
Description: Derive set.mm's original ax-c11n 34695 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 2138. Use aecom 2463 instead when this does not lengthen the proof. (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.)
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 2456 . . . . 5 (¬ ∀𝑦 𝑦 = 𝑥 → (𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧))
21com12 32 . . . 4 (𝑥 = 𝑧 → (¬ ∀𝑦 𝑦 = 𝑥 → ∀𝑦 𝑥 = 𝑧))
3 axc11r 2349 . . . . 5 (∀𝑥 𝑥 = 𝑦 → (∀𝑦 𝑥 = 𝑧 → ∀𝑥 𝑥 = 𝑧))
4 aev 2140 . . . . 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 2100 . . 3 𝑧 𝑥 = 𝑧
86, 7exlimiiv 2011 . 2 (∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑦 𝑦 = 𝑥 → ∀𝑦 𝑦 = 𝑥))
98pm2.18d 125 1 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-10 2174  ax-12 2203  ax-13 2408
This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1853  df-nf 1858
This theorem is referenced by:  aecom  2463  axi10  2748  wl-hbae1  33639  wl-ax11-lem3  33697  wl-ax11-lem8  33702  2sb5ndVD  39668  e2ebindVD  39670  e2ebindALT  39687  2sb5ndALT  39690
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