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Theorem axc11n 2428
Description: Derive set.mm's original ax-c11n 38869 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 2052. Use aecom 2429 instead when this does not lengthen the proof. Usage of this theorem is discouraged because it depends on ax-13 2374. (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 2382 . . . . 5 (¬ ∀𝑦 𝑦 = 𝑥 → (𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧))
21com12 32 . . . 4 (𝑥 = 𝑧 → (¬ ∀𝑦 𝑦 = 𝑥 → ∀𝑦 𝑥 = 𝑧))
3 axc11r 2368 . . . . 5 (∀𝑥 𝑥 = 𝑦 → (∀𝑦 𝑥 = 𝑧 → ∀𝑥 𝑥 = 𝑧))
4 aev 2054 . . . . 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 1928 . 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 1534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-10 2138  ax-12 2174  ax-13 2374
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1776  df-nf 1780
This theorem is referenced by:  aecom  2429  axi10  2702  wl-hbae1  37499  wl-ax11-lem3  37567  wl-ax11-lem8  37572  2sb5ndVD  44907  e2ebindVD  44909  e2ebindALT  44926  2sb5ndALT  44929
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