Theorem equvini 2140

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 1853 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 1845	. . . 4 |- ( z = x -> ( x = y -> z = y ) )
2		equequ2 1848	. . . . . 6 |- ( z = y -> ( x = z <-> x = y ) )
3	2	biimprd 226	. . . . 5 |- ( z = y -> ( x = y -> x = z ) )
4	3	anc2ri 560	. . . 4 |- ( z = y -> ( x = y -> ( x = z /\ z = y ) ) )
5	1, 4	syli 38	. . 3 |- ( z = x -> ( x = y -> ( x = z /\ z = y ) ) )
6		19.8a 1907	. . 3 |- ( ( x = z /\ z = y ) -> E. z ( x = z /\ z = y ) )
7	5, 6	syl6 34	. 2 |- ( z = x -> ( x = y -> E. z ( x = z /\ z = y ) ) )
8		ax13 2100	. . 3 |- ( -. z = x -> ( x = y -> A. z x = y ) )
9		ax6e 2055	. . . . 5 |- E. z z = y
10	9, 4	eximii 1704	. . . 4 |- E. z ( x = y -> ( x = z /\ z = y ) )
11	10	19.35i 1733	. . 3 |- ( A. z x = y -> E. z ( x = z /\ z = y ) )
12	8, 11	syl6 34	. 2 |- ( -. z = x -> ( x = y -> E. z ( x = z /\ z = y ) ) )
13	7, 12	pm2.61i 167	1 |- ( x = y -> E. z ( x = z /\ z = y ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 370   A.wal 1435   E.wex 1659

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-12 1904  ax-13 2052

This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1660  df-nf 1664

This theorem is referenced by:  2ax6elem  2242