Theorem equvini 2346

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 equvinv 1959 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 1948	. . . 4 |- ( z = x -> ( x = y -> z = y ) )
2		equeuclr 1950	. . . . 5 |- ( z = y -> ( x = y -> x = z ) )
3	2	anc2ri 581	. . . 4 |- ( z = y -> ( x = y -> ( x = z /\ z = y ) ) )
4	1, 3	syli 39	. . 3 |- ( z = x -> ( x = y -> ( x = z /\ z = y ) ) )
5		19.8a 2052	. . 3 |- ( ( x = z /\ z = y ) -> E. z ( x = z /\ z = y ) )
6	4, 5	syl6 35	. 2 |- ( z = x -> ( x = y -> E. z ( x = z /\ z = y ) ) )
7		ax13 2249	. . 3 |- ( -. z = x -> ( x = y -> A. z x = y ) )
8		ax6e 2250	. . . . 5 |- E. z z = y
9	8, 3	eximii 1764	. . . 4 |- E. z ( x = y -> ( x = z /\ z = y ) )
10	9	19.35i 1806	. . 3 |- ( A. z x = y -> E. z ( x = z /\ z = y ) )
11	7, 10	syl6 35	. 2 |- ( -. z = x -> ( x = y -> E. z ( x = z /\ z = y ) ) )
12	6, 11	pm2.61i 176	1 |- ( x = y -> E. z ( x = z /\ z = y ) )

Syntax hints:   -. wn 3   -> wi 4   /\ 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-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by:  2ax6elem  2449