Theorem equviniOLD

Description: Obsolete version of equvini as of 16-Sep-2023. (Contributed by NM, 10-Jan-1993) (Proof shortened by Andrew Salmon, 25-May-2011) (Proof shortened by Wolf Lammen, 15-Sep-2018) (New usage is discouraged.) (Proof modification is discouraged.)

	Ref	Expression
Assertion	equviniOLD	|- ( x = y -> E. z ( x = z /\ z = y ) )

Proof

Step	Hyp	Ref	Expression
1		equtr	|- ( z = x -> ( x = y -> z = y ) )
2		equeuclr	|- ( z = y -> ( x = y -> x = z ) )
3	2	anc2ri	|- ( z = y -> ( x = y -> ( x = z /\ z = y ) ) )
4	1 3	syli	|- ( z = x -> ( x = y -> ( x = z /\ z = y ) ) )
5		19.8a	|- ( ( x = z /\ z = y ) -> E. z ( x = z /\ z = y ) )
6	4 5	syl6	|- ( z = x -> ( x = y -> E. z ( x = z /\ z = y ) ) )
7		ax13	|- ( -. z = x -> ( x = y -> A. z x = y ) )
8		ax6e	|- E. z z = y
9	8 3	eximii	|- E. z ( x = y -> ( x = z /\ z = y ) )
10	9	19.35i	|- ( A. z x = y -> E. z ( x = z /\ z = y ) )
11	7 10	syl6	|- ( -. z = x -> ( x = y -> E. z ( x = z /\ z = y ) ) )
12	6 11	pm2.61i	|- ( x = y -> E. z ( x = z /\ z = y ) )