Theorem equvel 2347

Description: A variable elimination law for equality with no distinct variable requirements. Compare equvini 2346. (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
equvel	|- ( A. z ( z = x <-> z = y ) -> x = y )

Proof of Theorem equvel

Step	Hyp	Ref	Expression
1		albi 1746	. 2 |- ( A. z ( z = x <-> z = y ) -> ( A. z z = x <-> A. z z = y ) )
2		ax6e 2250	. . . 4 |- E. z z = y
3		biimpr 210	. . . . . 6 |- ( ( z = x <-> z = y ) -> ( z = y -> z = x ) )
4		ax7 1943	. . . . . 6 |- ( z = x -> ( z = y -> x = y ) )
5	3, 4	syli 39	. . . . 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 1764	. . 3 |- E. z ( ( z = x <-> z = y ) -> x = y )
8	7	19.35i 1806	. 2 |- ( A. z ( z = x <-> z = y ) -> E. z x = y )
9	4	spsd 2057	. . . . 5 |- ( z = x -> ( A. z z = y -> x = y ) )
10	9	sps 2055	. . . 4 |- ( A. z z = x -> ( A. z z = y -> x = y ) )
11	10	a1dd 50	. . 3 |- ( A. z z = x -> ( A. z z = y -> ( E. z x = y -> x = y ) ) )
12		nfeqf 2301	. . . . 5 |- ( ( -. A. z z = x /\ -. A. z z = y ) -> F/ z x = y )
13	12	19.9d 2070	. . . 4 |- ( ( -. A. z z = x /\ -. A. z z = y ) -> ( E. z x = y -> x = y ) )
14	13	ex 450	. . 3 |- ( -. A. z z = x -> ( -. A. z z = y -> ( E. z x = y -> x = y ) ) )
15	11, 14	bija 370	. 2 |- ( ( A. z z = x <-> A. z z = y ) -> ( E. z x = y -> x = y ) )
16	1, 8, 15	sylc 65	1 |- ( A. z ( z = x <-> z = y ) -> x = y )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   E.wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710

This theorem is referenced by: (None)