Theorem equvini

Description: A variable introduction law for equality. Lemma 15 of Monk2 p. 109, however we do not require z to be distinct from x and y . Usage of this theorem is discouraged because it depends on ax-13 . See equvinv for a shorter proof requiring fewer axioms when z is required to be distinct from x and y . (Contributed by NM, 10-Jan-1993) (Proof shortened by Andrew Salmon, 25-May-2011) (Proof shortened by Wolf Lammen, 16-Sep-2023) (New usage is discouraged.)

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

Proof

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