Metamath Proof Explorer


Theorem nd2

Description: A lemma for proving conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 1-Jan-2002) (New usage is discouraged.)

Ref Expression
Assertion nd2
|- ( A. x x = y -> -. A. x z e. y )

Proof

Step Hyp Ref Expression
1 elirrv
 |-  -. z e. z
2 stdpc4
 |-  ( A. y z e. y -> [ z / y ] z e. y )
3 1 nfnth
 |-  F/ y z e. z
4 elequ2
 |-  ( y = z -> ( z e. y <-> z e. z ) )
5 3 4 sbie
 |-  ( [ z / y ] z e. y <-> z e. z )
6 2 5 sylib
 |-  ( A. y z e. y -> z e. z )
7 1 6 mto
 |-  -. A. y z e. y
8 axc11
 |-  ( A. x x = y -> ( A. x z e. y -> A. y z e. y ) )
9 7 8 mtoi
 |-  ( A. x x = y -> -. A. x z e. y )