Theorem equvini 2060

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. See equvin 1753 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, 15-Sep-2018.)

Assertion

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

Proof of Theorem equvini

Step	Hyp	Ref	Expression
1		equtr 1745	. . . 4 |- ( z = x -> ( x = y -> z = y ) )
2		equequ2 1748	. . . . . 6 |- ( z = y -> ( x = z <-> x = y ) )
3	2	biimprd 223	. . . . 5 |- ( z = y -> ( x = y -> x = z ) )
4	3	anc2ri 558	. . . 4 |- ( z = y -> ( x = y -> ( x = z /\ z = y ) ) )
5	1, 4	syli 37	. . 3 |- ( z = x -> ( x = y -> ( x = z /\ z = y ) ) )
6		19.8a 1806	. . 3 |- ( ( x = z /\ z = y ) -> E. z ( x = z /\ z = y ) )
7	5, 6	syl6 33	. 2 |- ( z = x -> ( x = y -> E. z ( x = z /\ z = y ) ) )
8		ax13 2020	. . 3 |- ( -. z = x -> ( x = y -> A. z x = y ) )
9		ax6e 1971	. . . . 5 |- E. z z = y
10	9, 4	eximii 1637	. . . 4 |- E. z ( x = y -> ( x = z /\ z = y ) )
11	10	19.35i 1666	. . 3 |- ( A. z x = y -> E. z ( x = z /\ z = y ) )
12	8, 11	syl6 33	. 2 |- ( -. z = x -> ( x = y -> E. z ( x = z /\ z = y ) ) )
13	7, 12	pm2.61i 164	1 |- ( x = y -> E. z ( x = z /\ z = y ) )

Syntax hints:   -. wn 3   -> wi 4   /\ 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:  equvinOLD  2063  2ax6elem  2179