Theorem reuss2 3211

Description: Transfer uniqueness to a smaller subclass. (Contributed by NM, 20-Oct-2005.)

Assertion

Ref	Expression
reuss2	|- (((A C_ B /\ A.x e. A (ph -> ps)) /\ (E.x e. A ph /\ E!x e. B ps)) -> E!x e. A ph)

Distinct variable groups:   x,A   x,B

Allowed substitution hints:   ph(x)   ps(x)

Proof of Theorem reuss2

Step	Hyp	Ref	Expression
1		df-rex 2306	. . 3 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
2		df-reu 2307	. . 3 |- (E!x e. B ps <-> E!x(x e. B /\ ps))
3	1, 2	anbi12i 433	. 2 |- ((E.x e. A ph /\ E!x e. B ps) <-> (E.x(x e. A /\ ph) /\ E!x(x e. B /\ ps)))
4		df-ral 2305	. . . . . . 7 |- (A.x e. A (ph -> ps) <-> A.x(x e. A -> (ph -> ps)))
5		ssel 2933	. . . . . . . . . . . . . 14 |- (A C_ B -> (x e. A -> x e. B))
6		prth 326	. . . . . . . . . . . . . 14 |- (((x e. A -> x e. B) /\ (ph -> ps)) -> ((x e. A /\ ph) -> (x e. B /\ ps)))
7	5, 6	sylan 267	. . . . . . . . . . . . 13 |- ((A C_ B /\ (ph -> ps)) -> ((x e. A /\ ph) -> (x e. B /\ ps)))
8	7	exp4b 349	. . . . . . . . . . . 12 |- (A C_ B -> ((ph -> ps) -> (x e. A -> (ph -> (x e. B /\ ps)))))
9	8	com23 72	. . . . . . . . . . 11 |- (A C_ B -> (x e. A -> ((ph -> ps) -> (ph -> (x e. B /\ ps)))))
10	9	a2d 23	. . . . . . . . . 10 |- (A C_ B -> ((x e. A -> (ph -> ps)) -> (x e. A -> (ph -> (x e. B /\ ps)))))
11	10	imp4a 331	. . . . . . . . 9 |- (A C_ B -> ((x e. A -> (ph -> ps)) -> ((x e. A /\ ph) -> (x e. B /\ ps))))
12	11	alimdv 1756	. . . . . . . 8 |- (A C_ B -> (A.x(x e. A -> (ph -> ps)) -> A.x((x e. A /\ ph) -> (x e. B /\ ps))))
13	12	imp 115	. . . . . . 7 |- ((A C_ B /\ A.x(x e. A -> (ph -> ps))) -> A.x((x e. A /\ ph) -> (x e. B /\ ps)))
14	4, 13	sylan2b 271	. . . . . 6 |- ((A C_ B /\ A.x e. A (ph -> ps)) -> A.x((x e. A /\ ph) -> (x e. B /\ ps)))
15		euimmo 1964	. . . . . 6 |- (A.x((x e. A /\ ph) -> (x e. B /\ ps)) -> (E!x(x e. B /\ ps) -> E*x(x e. A /\ ph)))
16	14, 15	syl 14	. . . . 5 |- ((A C_ B /\ A.x e. A (ph -> ps)) -> (E!x(x e. B /\ ps) -> E*x(x e. A /\ ph)))
17		eu5 1944	. . . . . 6 |- (E!x(x e. A /\ ph) <-> (E.x(x e. A /\ ph) /\ E*x(x e. A /\ ph)))
18	17	simplbi2 367	. . . . 5 |- (E.x(x e. A /\ ph) -> (E*x(x e. A /\ ph) -> E!x(x e. A /\ ph)))
19	16, 18	syl9 66	. . . 4 |- ((A C_ B /\ A.x e. A (ph -> ps)) -> (E.x(x e. A /\ ph) -> (E!x(x e. B /\ ps) -> E!x(x e. A /\ ph))))
20	19	imp32 244	. . 3 |- (((A C_ B /\ A.x e. A (ph -> ps)) /\ (E.x(x e. A /\ ph) /\ E!x(x e. B /\ ps))) -> E!x(x e. A /\ ph))
21		df-reu 2307	. . 3 |- (E!x e. A ph <-> E!x(x e. A /\ ph))
22	20, 21	sylibr 137	. 2 |- (((A C_ B /\ A.x e. A (ph -> ps)) /\ (E.x(x e. A /\ ph) /\ E!x(x e. B /\ ps))) -> E!x e. A ph)
23	3, 22	sylan2b 271	1 |- (((A C_ B /\ A.x e. A (ph -> ps)) /\ (E.x e. A ph /\ E!x e. B ps)) -> E!x e. A ph)

Syntax hints:   -> wi 4   /\ wa 97  A.wal 1240  E.wex 1378   e. wcel 1390  E!weu 1897  E*wmo 1898  A.wral 2300  E.wrex 2301  E!wreu 2302   C_ wss 2911

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-ral 2305  df-rex 2306  df-reu 2307  df-in 2918  df-ss 2925

This theorem is referenced by:  reuss  3212  reuun1  3213  riotass2  5437