Theorem axc11n 2424

Description: Derive set.mm's original ax-c11n 38222 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 2057. Use aecom 2425 instead when this does not lengthen the proof. Usage of this theorem is discouraged because it depends on ax-13 2370. (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 2378	. . . . 5 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧))
2	1	com12 32	. . . 4 ⊢ (𝑥 = 𝑧 → (¬ ∀𝑦 𝑦 = 𝑥 → ∀𝑦 𝑥 = 𝑧))
3		axc11r 2364	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦 𝑥 = 𝑧 → ∀𝑥 𝑥 = 𝑧))
4		aev 2059	. . . . 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 2017	. . 3 ⊢ ∃𝑧 𝑥 = 𝑧
8	6, 7	exlimiiv 1933	. 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 1538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-10 2136  ax-12 2170  ax-13 2370

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1781  df-nf 1785

This theorem is referenced by:  aecom  2425  axi10  2699  wl-hbae1  36852  wl-ax11-lem3  36913  wl-ax11-lem8  36918  2sb5ndVD  44134  e2ebindVD  44136  e2ebindALT  44153  2sb5ndALT  44156