Theorem mo23 2040

Description: An implication between two definitions of "there exists at most one." (Contributed by Jim Kingdon, 25-Jun-2018.)

Hypothesis

Ref	Expression
mo23.1	|- F/ y ph

Assertion

Ref	Expression
mo23	|- ( E. y A. x ( ph -> x = y ) -> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem mo23

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mo23.1	. . . . 5 |- F/ y ph
2		nfv 1508	. . . . 5 |- F/ y x = z
3	1, 2	nfim 1551	. . . 4 |- F/ y ( ph -> x = z )
4	3	nfal 1555	. . 3 |- F/ y A. x ( ph -> x = z )
5		nfv 1508	. . 3 |- F/ z A. x ( ph -> x = y )
6		equequ2 1689	. . . . 5 |- ( z = y -> ( x = z <-> x = y ) )
7	6	imbi2d 229	. . . 4 |- ( z = y -> ( ( ph -> x = z ) <-> ( ph -> x = y ) ) )
8	7	albidv 1796	. . 3 |- ( z = y -> ( A. x ( ph -> x = z ) <-> A. x ( ph -> x = y ) ) )
9	4, 5, 8	cbvex 1729	. 2 |- ( E. z A. x ( ph -> x = z ) <-> E. y A. x ( ph -> x = y ) )
10		nfs1v 1912	. . . . . . . 8 |- F/ x [ y / x ] ph
11		nfv 1508	. . . . . . . 8 |- F/ x y = z
12	10, 11	nfim 1551	. . . . . . 7 |- F/ x ( [ y / x ] ph -> y = z )
13		sbequ2 1742	. . . . . . . 8 |- ( x = y -> ( [ y / x ] ph -> ph ) )
14		ax-8 1482	. . . . . . . 8 |- ( x = y -> ( x = z -> y = z ) )
15	13, 14	imim12d 74	. . . . . . 7 |- ( x = y -> ( ( ph -> x = z ) -> ( [ y / x ] ph -> y = z ) ) )
16	3, 12, 15	cbv3 1720	. . . . . 6 |- ( A. x ( ph -> x = z ) -> A. y ( [ y / x ] ph -> y = z ) )
17	16	ancli 321	. . . . 5 |- ( A. x ( ph -> x = z ) -> ( A. x ( ph -> x = z ) /\ A. y ( [ y / x ] ph -> y = z ) ) )
18	3	nfri 1499	. . . . . 6 |- ( ( ph -> x = z ) -> A. y ( ph -> x = z ) )
19	12	nfri 1499	. . . . . 6 |- ( ( [ y / x ] ph -> y = z ) -> A. x ( [ y / x ] ph -> y = z ) )
20	18, 19	aaanh 1565	. . . . 5 |- ( A. x A. y ( ( ph -> x = z ) /\ ( [ y / x ] ph -> y = z ) ) <-> ( A. x ( ph -> x = z ) /\ A. y ( [ y / x ] ph -> y = z ) ) )
21	17, 20	sylibr 133	. . . 4 |- ( A. x ( ph -> x = z ) -> A. x A. y ( ( ph -> x = z ) /\ ( [ y / x ] ph -> y = z ) ) )
22		anim12 341	. . . . . 6 |- ( ( ( ph -> x = z ) /\ ( [ y / x ] ph -> y = z ) ) -> ( ( ph /\ [ y / x ] ph ) -> ( x = z /\ y = z ) ) )
23		equtr2 1687	. . . . . 6 |- ( ( x = z /\ y = z ) -> x = y )
24	22, 23	syl6 33	. . . . 5 |- ( ( ( ph -> x = z ) /\ ( [ y / x ] ph -> y = z ) ) -> ( ( ph /\ [ y / x ] ph ) -> x = y ) )
25	24	2alimi 1432	. . . 4 |- ( A. x A. y ( ( ph -> x = z ) /\ ( [ y / x ] ph -> y = z ) ) -> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
26	21, 25	syl 14	. . 3 |- ( A. x ( ph -> x = z ) -> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
27	26	exlimiv 1577	. 2 |- ( E. z A. x ( ph -> x = z ) -> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
28	9, 27	sylbir 134	1 |- ( E. y A. x ( ph -> x = y ) -> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1329   F/wnf 1436   E.wex 1468   [wsb 1735

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736

This theorem is referenced by:  modc  2042  eu2  2043  eu3h  2044