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Theorem axc11n 2191
Description: Derive set.mm's original ax-c11n 32693 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). Use aecom 2194 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.)
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 equcomi 1893 . . . . 5 (𝑧 = 𝑥𝑥 = 𝑧)
2 dveeq1 2185 . . . . 5 (¬ ∀𝑦 𝑦 = 𝑥 → (𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧))
31, 2syl5com 31 . . . 4 (𝑧 = 𝑥 → (¬ ∀𝑦 𝑦 = 𝑥 → ∀𝑦 𝑥 = 𝑧))
4 axc11r 2073 . . . . 5 (∀𝑥 𝑥 = 𝑦 → (∀𝑦 𝑥 = 𝑧 → ∀𝑥 𝑥 = 𝑧))
5 aev 1930 . . . . 5 (∀𝑥 𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑥)
64, 5syl6 34 . . . 4 (∀𝑥 𝑥 = 𝑦 → (∀𝑦 𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑥))
73, 6syl9 74 . . 3 (𝑧 = 𝑥 → (∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑦 𝑦 = 𝑥 → ∀𝑦 𝑦 = 𝑥)))
8 ax6ev 1838 . . 3 𝑧 𝑧 = 𝑥
97, 8exlimiiv 1808 . 2 (∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑦 𝑦 = 𝑥 → ∀𝑦 𝑦 = 𝑥))
109pm2.18d 117 1 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-10 1965  ax-12 1983  ax-13 2137
This theorem depends on definitions:  df-bi 192  df-an 380  df-ex 1693  df-nf 1697
This theorem is referenced by:  aecom  2194  axi10  2482  wl-ax11-lem3  32142  wl-ax11-lem8  32147  2sb5ndVD  37655  e2ebindVD  37657  e2ebindALT  37674  2sb5ndALT  37677
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