Theorem equvini 1641

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 1403	. 2 |- ( A. z z = x \/ ( A. z z = y \/ A. z ( x = y -> A. z x = y ) ) )
2		equcomi 1592	. . . . . . 7 |- ( z = x -> x = z )
3	2	alimi 1344	. . . . . 6 |- ( A. z z = x -> A. z x = z )
4		a9e 1586	. . . . . 6 |- E. z z = y
5	3, 4	jctir 296	. . . . 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 1511	. . . 4 |- ( ( A. z x = z /\ E. z z = y ) -> E. z ( x = z /\ z = y ) )
8	6, 7	syl6 29	. . 3 |- ( A. z z = x -> ( x = y -> E. z ( x = z /\ z = y ) ) )
9		a9e 1586	. . . . . . . 8 |- E. z z = x
10	2	eximi 1491	. . . . . . . 8 |- ( E. z z = x -> E. z x = z )
11	9, 10	ax-mp 7	. . . . . . 7 |- E. z x = z
12	11	2a1i 24	. . . . . 6 |- ( A. z z = y -> ( x = y -> E. z x = z ) )
13	12	anc2ri 313	. . . . 5 |- ( A. z z = y -> ( x = y -> ( E. z x = z /\ A. z z = y ) ) )
14		19.29r 1512	. . . . 5 |- ( ( E. z x = z /\ A. z z = y ) -> E. z ( x = z /\ z = y ) )
15	13, 14	syl6 29	. . . 4 |- ( A. z z = y -> ( x = y -> E. z ( x = z /\ z = y ) ) )
16		ax-8 1395	. . . . . . . . . . . 12 |- ( x = z -> ( x = y -> z = y ) )
17	16	anc2li 312	. . . . . . . . . . 11 |- ( x = z -> ( x = y -> ( x = z /\ z = y ) ) )
18	17	equcoms 1594	. . . . . . . . . 10 |- ( z = x -> ( x = y -> ( x = z /\ z = y ) ) )
19	18	com12 27	. . . . . . . . 9 |- ( x = y -> ( z = x -> ( x = z /\ z = y ) ) )
20	19	alimi 1344	. . . . . . . 8 |- ( A. z x = y -> A. z ( z = x -> ( x = z /\ z = y ) ) )
21		exim 1490	. . . . . . . 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 1430	. . . 4 |- ( A. z ( x = y -> A. z x = y ) -> ( x = y -> E. z ( x = z /\ z = y ) ) )
26	15, 25	jaoi 636	. . 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 636	. 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 7	1 |- ( x = y -> E. z ( x = z /\ z = y ) )

Syntax hints:   -> wi 4   /\ wa 97   \/ wo 629   A.wal 1241   = wceq 1243   E.wex 1381

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-i12 1398  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  sbequi  1720  equvin  1743