Theorem equvel

Description: A variable elimination law for equality with no distinct variable requirements. Compare equvini . Usage of this theorem is discouraged because it depends on ax-13 . Use equvelv when possible. (Contributed by NM, 1-Mar-2013) (Proof shortened by Mario Carneiro, 17-Oct-2016) (Proof shortened by Wolf Lammen, 15-Jun-2019) (New usage is discouraged.)

	Ref	Expression
Assertion	equvel	|- ( A. z ( z = x <-> z = y ) -> x = y )

Proof

Step	Hyp	Ref	Expression
1		albi	|- ( A. z ( z = x <-> z = y ) -> ( A. z z = x <-> A. z z = y ) )
2		ax6e	|- E. z z = y
3		biimpr	|- ( ( z = x <-> z = y ) -> ( z = y -> z = x ) )
4		ax7	|- ( z = x -> ( z = y -> x = y ) )
5	3 4	syli	|- ( ( z = x <-> z = y ) -> ( z = y -> x = y ) )
6	5	com12	|- ( z = y -> ( ( z = x <-> z = y ) -> x = y ) )
7	2 6	eximii	|- E. z ( ( z = x <-> z = y ) -> x = y )
8	7	19.35i	|- ( A. z ( z = x <-> z = y ) -> E. z x = y )
9	4	spsd	|- ( z = x -> ( A. z z = y -> x = y ) )
10	9	sps	|- ( A. z z = x -> ( A. z z = y -> x = y ) )
11	10	a1dd	|- ( A. z z = x -> ( A. z z = y -> ( E. z x = y -> x = y ) ) )
12		nfeqf	|- ( ( -. A. z z = x /\ -. A. z z = y ) -> F/ z x = y )
13	12	19.9d	|- ( ( -. A. z z = x /\ -. A. z z = y ) -> ( E. z x = y -> x = y ) )
14	13	ex	|- ( -. A. z z = x -> ( -. A. z z = y -> ( E. z x = y -> x = y ) ) )
15	11 14	bija	|- ( ( A. z z = x <-> A. z z = y ) -> ( E. z x = y -> x = y ) )
16	1 8 15	sylc	|- ( A. z ( z = x <-> z = y ) -> x = y )