Theorem axc11n 2307

Description: Derive set.mm's original ax-c11n 34173 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 1981. Use aecom 2311 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	|- ( A. x x = y -> A. y y = x )

Proof of Theorem axc11n

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dveeq1 2300	. . . . 5 |- ( -. A. y y = x -> ( x = z -> A. y x = z ) )
2	1	com12 32	. . . 4 |- ( x = z -> ( -. A. y y = x -> A. y x = z ) )
3		axc11r 2187	. . . . 5 |- ( A. x x = y -> ( A. y x = z -> A. x x = z ) )
4		aev 1983	. . . . 5 |- ( A. x x = z -> A. y y = x )
5	3, 4	syl6 35	. . . 4 |- ( A. x x = y -> ( A. y x = z -> A. y y = x ) )
6	2, 5	syl9 77	. . 3 |- ( x = z -> ( A. x x = y -> ( -. A. y y = x -> A. y y = x ) ) )
7		ax6evr 1942	. . 3 |- E. z x = z
8	6, 7	exlimiiv 1859	. 2 |- ( A. x x = y -> ( -. A. y y = x -> A. y y = x ) )
9	8	pm2.18d 124	1 |- ( A. x x = y -> A. y y = x )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by:  aecom  2311  axi10  2599  wl-hbae1  33303  wl-ax11-lem3  33364  wl-ax11-lem8  33369  2sb5ndVD  39146  e2ebindVD  39148  e2ebindALT  39165  2sb5ndALT  39168