Theorem equveli 2061

Description: A variable elimination law for equality with no distinct variable requirements. (Compare equvini 2060.) (Contributed by NM, 1-Mar-2013.) (Proof shortened by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 15-Jun-2019.)

Assertion

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

Proof of Theorem equveli

Step	Hyp	Ref	Expression
1		ax6e 1971	. . . 4 |- E. z z = y
2		bi2 198	. . . . . 6 |- ( ( z = x <-> z = y ) -> ( z = y -> z = x ) )
3		ax-7 1739	. . . . . 6 |- ( z = x -> ( z = y -> x = y ) )
4	2, 3	syli 37	. . . . 5 |- ( ( z = x <-> z = y ) -> ( z = y -> x = y ) )
5	4	com12 31	. . . 4 |- ( z = y -> ( ( z = x <-> z = y ) -> x = y ) )
6	1, 5	eximii 1637	. . 3 |- E. z ( ( z = x <-> z = y ) -> x = y )
7	6	19.35i 1666	. 2 |- ( A. z ( z = x <-> z = y ) -> E. z x = y )
8		albi 1619	. . 3 |- ( A. z ( z = x <-> z = y ) -> ( A. z z = x <-> A. z z = y ) )
9	3	spsd 1816	. . . . . 6 |- ( z = x -> ( A. z z = y -> x = y ) )
10	9	sps 1814	. . . . 5 |- ( A. z z = x -> ( A. z z = y -> x = y ) )
11	10	a1dd 46	. . . 4 |- ( A. z z = x -> ( A. z z = y -> ( E. z x = y -> x = y ) ) )
12		nfeqf 2018	. . . . . 6 |- ( ( -. A. z z = x /\ -. A. z z = y ) -> F/ z x = y )
13	12	19.9d 1840	. . . . 5 |- ( ( -. A. z z = x /\ -. A. z z = y ) -> ( E. z x = y -> x = y ) )
14	13	ex 434	. . . 4 |- ( -. A. z z = x -> ( -. A. z z = y -> ( E. z x = y -> x = y ) ) )
15	11, 14	bija 355	. . 3 |- ( ( A. z z = x <-> A. z z = y ) -> ( E. z x = y -> x = y ) )
16	8, 15	syl 16	. 2 |- ( A. z ( z = x <-> z = y ) -> ( E. z x = y -> x = y ) )
17	7, 16	mpd 15	1 |- ( A. z ( z = x <-> z = y ) -> x = y )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   A.wal 1377   E.wex 1596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-12 1803  ax-13 1968

This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1597  df-nf 1600

This theorem is referenced by: (None)