Theorem equvini 1804

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 (making the proof longer). (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Assertion

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

Proof of Theorem equvini

Step	Hyp	Ref	Expression
1		ax12or 1554	. 2 |- ( A. z z = x \/ ( A. z z = y \/ A. z ( x = y -> A. z x = y ) ) )
2		equcomi 1750	. . . . . . 7 |- ( z = x -> x = z )
3	2	alimi 1501	. . . . . 6 |- ( A. z z = x -> A. z x = z )
4		a9e 1742	. . . . . 6 |- E. z z = y
5	3, 4	jctir 313	. . . . 5 |- ( A. z z = x -> ( A. z x = z /\ E. z z = y ) )
6	5	a1d 22	. . . 4 |- ( A. z z = x -> ( x = y -> ( A. z x = z /\ E. z z = y ) ) )
7		19.29 1666	. . . 4 |- ( ( A. z x = z /\ E. z z = y ) -> E. z ( x = z /\ z = y ) )
8	6, 7	syl6 33	. . 3 |- ( A. z z = x -> ( x = y -> E. z ( x = z /\ z = y ) ) )
9		a9e 1742	. . . . . . . 8 |- E. z z = x
10	2	eximi 1646	. . . . . . . 8 |- ( E. z z = x -> E. z x = z )
11	9, 10	ax-mp 5	. . . . . . 7 |- E. z x = z
12	11	2a1i 27	. . . . . 6 |- ( A. z z = y -> ( x = y -> E. z x = z ) )
13	12	anc2ri 330	. . . . 5 |- ( A. z z = y -> ( x = y -> ( E. z x = z /\ A. z z = y ) ) )
14		19.29r 1667	. . . . 5 |- ( ( E. z x = z /\ A. z z = y ) -> E. z ( x = z /\ z = y ) )
15	13, 14	syl6 33	. . . 4 |- ( A. z z = y -> ( x = y -> E. z ( x = z /\ z = y ) ) )
16		ax-8 1550	. . . . . . . . . . . 12 |- ( x = z -> ( x = y -> z = y ) )
17	16	anc2li 329	. . . . . . . . . . 11 |- ( x = z -> ( x = y -> ( x = z /\ z = y ) ) )
18	17	equcoms 1754	. . . . . . . . . 10 |- ( z = x -> ( x = y -> ( x = z /\ z = y ) ) )
19	18	com12 30	. . . . . . . . 9 |- ( x = y -> ( z = x -> ( x = z /\ z = y ) ) )
20	19	alimi 1501	. . . . . . . 8 |- ( A. z x = y -> A. z ( z = x -> ( x = z /\ z = y ) ) )
21		exim 1645	. . . . . . . 8 |- ( A. z ( z = x -> ( x = z /\ z = y ) ) -> ( E. z z = x -> E. z ( x = z /\ z = y ) ) )
22	20, 21	syl 14	. . . . . . 7 |- ( A. z x = y -> ( E. z z = x -> E. z ( x = z /\ z = y ) ) )
23	9, 22	mpi 15	. . . . . 6 |- ( A. z x = y -> E. z ( x = z /\ z = y ) )
24	23	imim2i 12	. . . . 5 |- ( ( x = y -> A. z x = y ) -> ( x = y -> E. z ( x = z /\ z = y ) ) )
25	24	sps 1583	. . . 4 |- ( A. z ( x = y -> A. z x = y ) -> ( x = y -> E. z ( x = z /\ z = y ) ) )
26	15, 25	jaoi 721	. . 3 |- ( ( A. z z = y \/ A. z ( x = y -> A. z x = y ) ) -> ( x = y -> E. z ( x = z /\ z = y ) ) )
27	8, 26	jaoi 721	. 2 |- ( ( A. z z = x \/ ( A. z z = y \/ A. z ( x = y -> A. z x = y ) ) ) -> ( x = y -> E. z ( x = z /\ z = y ) ) )
28	1, 27	ax-mp 5	1 |- ( x = y -> E. z ( x = z /\ z = y ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 713   A.wal 1393   = wceq 1395   E.wex 1538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-i12 1553  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  sbequi  1885  equvin  1909