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.

Step	Hyp	Ref	Expression
1		dveeq1 2375	. . . . 5 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧))
2	1	com12 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)
5	3, 4	syl6 35	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦 𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑥))
6	2, 5	syl9 77	. . 3 ⊢ (𝑥 = 𝑧 → (∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑦 𝑦 = 𝑥 → ∀𝑦 𝑦 = 𝑥)))
7		ax6evr 2011	. . 3 ⊢ ∃𝑧 𝑥 = 𝑧
8	6, 7	exlimiiv 1927	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑦 𝑦 = 𝑥 → ∀𝑦 𝑦 = 𝑥))
9	8	pm2.18d 127	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

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