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Metamath Proof Explorer

Theorem axc11n

Description: Derive set.mm's original ax-c11n 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 . Use aecom instead when this does not lengthen the proof. Usage of this theorem is discouraged because it depends on ax-13 . (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.)

Ref Expression
Assertion axc11n 
|- ( A. x x = y -> A. y y = x )

Proof

Step Hyp Ref Expression
1 dveeq1 
 |-  ( -. A. y y = x -> ( x = z -> A. y x = z ) )
2 1 com12 
 |-  ( x = z -> ( -. A. y y = x -> A. y x = z ) )
3 axc11r 
 |-  ( A. x x = y -> ( A. y x = z -> A. x x = z ) )
4 aev 
 |-  ( A. x x = z -> A. y y = x )
5 3 4 syl6 
 |-  ( A. x x = y -> ( A. y x = z -> A. y y = x ) )
6 2 5 syl9 
 |-  ( x = z -> ( A. x x = y -> ( -. A. y y = x -> A. y y = x ) ) )
7 ax6evr 
 |-  E. z x = z
8 6 7 exlimiiv 
 |-  ( A. x x = y -> ( -. A. y y = x -> A. y y = x ) )
9 8 pm2.18d 
 |-  ( A. x x = y -> A. y y = x )