Theorem mo23 1938

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/yph

Assertion

Ref	Expression
mo23	|- (E.yA.x(ph -> x = y) -> A.xA.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/yph
2		nfv 1418	. . . . 5 |- F/y x = z
3	1, 2	nfim 1461	. . . 4 |- F/y(ph -> x = z)
4	3	nfal 1465	. . 3 |- F/yA.x(ph -> x = z)
5		nfv 1418	. . 3 |- F/zA.x(ph -> x = y)
6		equequ2 1596	. . . . 5 |- (z = y -> (x = z <-> x = y))
7	6	imbi2d 219	. . . 4 |- (z = y -> ((ph -> x = z) <-> (ph -> x = y)))
8	7	albidv 1702	. . 3 |- (z = y -> (A.x(ph -> x = z) <-> A.x(ph -> x = y)))
9	4, 5, 8	cbvex 1636	. 2 |- (E.zA.x(ph -> x = z) <-> E.yA.x(ph -> x = y))
10		nfs1v 1812	. . . . . . . 8 |- F/x[y / x]ph
11		nfv 1418	. . . . . . . 8 |- F/x y = z
12	10, 11	nfim 1461	. . . . . . 7 |- F/x([y / x]ph -> y = z)
13		sbequ2 1649	. . . . . . . 8 |- (x = y -> ([y / x]ph -> ph))
14		ax-8 1392	. . . . . . . 8 |- (x = y -> (x = z -> y = z))
15	13, 14	imim12d 68	. . . . . . 7 |- (x = y -> ((ph -> x = z) -> ([y / x]ph -> y = z)))
16	3, 12, 15	cbv3 1627	. . . . . 6 |- (A.x(ph -> x = z) -> A.y([y / x]ph -> y = z))
17	16	ancli 306	. . . . 5 |- (A.x(ph -> x = z) -> (A.x(ph -> x = z) /\ A.y([y / x]ph -> y = z)))
18	3	nfri 1409	. . . . . 6 |- ((ph -> x = z) -> A.y(ph -> x = z))
19	12	nfri 1409	. . . . . 6 |- (([y / x]ph -> y = z) -> A.x([y / x]ph -> y = z))
20	18, 19	aaanh 1475	. . . . 5 |- (A.xA.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 137	. . . 4 |- (A.x(ph -> x = z) -> A.xA.y((ph -> x = z) /\ ([y / x]ph -> y = z)))
22		prth 326	. . . . . 6 |- (((ph -> x = z) /\ ([y / x]ph -> y = z)) -> ((ph /\ [y / x]ph) -> (x = z /\ y = z)))
23		equtr2 1594	. . . . . 6 |- ((x = z /\ y = z) -> x = y)
24	22, 23	syl6 29	. . . . 5 |- (((ph -> x = z) /\ ([y / x]ph -> y = z)) -> ((ph /\ [y / x]ph) -> x = y))
25	24	2alimi 1342	. . . 4 |- (A.xA.y((ph -> x = z) /\ ([y / x]ph -> y = z)) -> A.xA.y((ph /\ [y / x]ph) -> x = y))
26	21, 25	syl 14	. . 3 |- (A.x(ph -> x = z) -> A.xA.y((ph /\ [y / x]ph) -> x = y))
27	26	exlimiv 1486	. 2 |- (E.zA.x(ph -> x = z) -> A.xA.y((ph /\ [y / x]ph) -> x = y))
28	9, 27	sylbir 125	1 |- (E.yA.x(ph -> x = y) -> A.xA.y((ph /\ [y / x]ph) -> x = y))

Syntax hints:   -> wi 4   /\ wa 97  A.wal 1240   F/wnf 1346  E.wex 1378  [wsb 1642

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-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643

This theorem is referenced by:  modc  1940  eu2  1941  eu3h  1942