Theorem equveli 2180

Description: A variable elimination law for equality with no distinct variable requirements. Compare equvini 2179. (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		albi 1690	. 2 |- ( A. z ( z = x <-> z = y ) -> ( A. z z = x <-> A. z z = y ) )
2		ax6e 2094	. . . 4 |- E. z z = y
3		biimpr 202	. . . . . 6 |- ( ( z = x <-> z = y ) -> ( z = y -> z = x ) )
4		ax7 1860	. . . . . 6 |- ( z = x -> ( z = y -> x = y ) )
5	3, 4	syli 38	. . . . 5 |- ( ( z = x <-> z = y ) -> ( z = y -> x = y ) )
6	5	com12 32	. . . 4 |- ( z = y -> ( ( z = x <-> z = y ) -> x = y ) )
7	2, 6	eximii 1709	. . 3 |- E. z ( ( z = x <-> z = y ) -> x = y )
8	7	19.35i 1741	. 2 |- ( A. z ( z = x <-> z = y ) -> E. z x = y )
9	4	spsd 1945	. . . . 5 |- ( z = x -> ( A. z z = y -> x = y ) )
10	9	sps 1943	. . . 4 |- ( A. z z = x -> ( A. z z = y -> x = y ) )
11	10	a1dd 47	. . 3 |- ( A. z z = x -> ( A. z z = y -> ( E. z x = y -> x = y ) ) )
12		nfeqf 2139	. . . . 5 |- ( ( -. A. z z = x /\ -. A. z z = y ) -> F/ z x = y )
13	12	19.9d 1968	. . . 4 |- ( ( -. A. z z = x /\ -. A. z z = y ) -> ( E. z x = y -> x = y ) )
14	13	ex 436	. . 3 |- ( -. A. z z = x -> ( -. A. z z = y -> ( E. z x = y -> x = y ) ) )
15	11, 14	bija 357	. 2 |- ( ( A. z z = x <-> A. z z = y ) -> ( E. z x = y -> x = y ) )
16	1, 8, 15	sylc 62	1 |- ( A. z ( z = x <-> z = y ) -> x = y )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   /\ wa 371   A.wal 1442   E.wex 1663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-12 1933  ax-13 2091

This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1664  df-nf 1668

This theorem is referenced by: (None)