Theorem reuss2 2327

Description: Transfer uniqueness to a smaller subclass.

Assertion

Ref	Expression
reuss2	|- (((A (_ 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

Proof of Theorem reuss2

Step	Hyp	Ref	Expression
1		prth 559	. . . . . . . . . . . . . 14 |- (((x e. A -> x e. B) /\ (ph -> ps)) -> ((x e. A /\ ph) -> (x e. B /\ ps)))
2		ssel 2115	. . . . . . . . . . . . . 14 |- (A (_ B -> (x e. A -> x e. B))
3	1, 2	sylan 450	. . . . . . . . . . . . 13 |- ((A (_ B /\ (ph -> ps)) -> ((x e. A /\ ph) -> (x e. B /\ ps)))
4	3	exp4b 379	. . . . . . . . . . . 12 |- (A (_ B -> ((ph -> ps) -> (x e. A -> (ph -> (x e. B /\ ps)))))
5	4	com23 32	. . . . . . . . . . 11 |- (A (_ B -> (x e. A -> ((ph -> ps) -> (ph -> (x e. B /\ ps)))))
6	5	a2d 13	. . . . . . . . . 10 |- (A (_ B -> ((x e. A -> (ph -> ps)) -> (x e. A -> (ph -> (x e. B /\ ps)))))
7	6	imp4a 362	. . . . . . . . 9 |- (A (_ B -> ((x e. A -> (ph -> ps)) -> ((x e. A /\ ph) -> (x e. B /\ ps))))
8	7	19.20dv 1327	. . . . . . . 8 |- (A (_ B -> (A.x(x e. A -> (ph -> ps)) -> A.x((x e. A /\ ph) -> (x e. B /\ ps))))
9	8	imp 348	. . . . . . 7 |- ((A (_ B /\ A.x(x e. A -> (ph -> ps))) -> A.x((x e. A /\ ph) -> (x e. B /\ ps)))
10		df-ral 1695	. . . . . . 7 |- (A.x e. A (ph -> ps) <-> A.x(x e. A -> (ph -> ps)))
11	9, 10	sylan2b 454	. . . . . 6 |- ((A (_ B /\ A.x e. A (ph -> ps)) -> A.x((x e. A /\ ph) -> (x e. B /\ ps)))
12		euimmo 1459	. . . . . 6 |- (A.x((x e. A /\ ph) -> (x e. B /\ ps)) -> (E!x(x e. B /\ ps) -> E*x(x e. A /\ ph)))
13	11, 12	syl 10	. . . . 5 |- ((A (_ B /\ A.x e. A (ph -> ps)) -> (E!x(x e. B /\ ps) -> E*x(x e. A /\ ph)))
14		eu5 1448	. . . . . . 7 |- (E!x(x e. A /\ ph) <-> (E.x(x e. A /\ ph) /\ E*x(x e. A /\ ph)))
15	14	biimpri 150	. . . . . 6 |- ((E.x(x e. A /\ ph) /\ E*x(x e. A /\ ph)) -> E!x(x e. A /\ ph))
16	15	ex 371	. . . . 5 |- (E.x(x e. A /\ ph) -> (E*x(x e. A /\ ph) -> E!x(x e. A /\ ph)))
17	13, 16	syl9 57	. . . 4 |- ((A (_ 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))))
18	17	imp32 361	. . 3 |- (((A (_ 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))
19		df-reu 1697	. . 3 |- (E!x e. A ph <-> E!x(x e. A /\ ph))
20	18, 19	sylibr 198	. 2 |- (((A (_ B /\ A.x e. A (ph -> ps)) /\ (E.x(x e. A /\ ph) /\ E!x(x e. B /\ ps))) -> E!x e. A ph)
21		df-rex 1696	. . 3 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
22		df-reu 1697	. . 3 |- (E!x e. B ps <-> E!x(x e. B /\ ps))
23	21, 22	anbi12i 485	. 2 |- ((E.x e. A ph /\ E!x e. B ps) <-> (E.x(x e. A /\ ph) /\ E!x(x e. B /\ ps)))
24	20, 23	sylan2b 454	1 |- (((A (_ 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 3   /\ wa 221  A.wal 990   e. wcel 994  E.wex 1016  E!weu 1419  E*wmo 1420  A.wral 1691  E.wrex 1692  E!wreu 1693   (_ wss 2099

This theorem is referenced by:  reuss 2328  reuun1 2329  reuuniss2 3114  grpidinv2 8277  grpinv 8286

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-10 1002  ax-11 1003  ax-12 1004  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ral 1695  df-rex 1696  df-reu 1697  df-in 2103  df-ss 2105

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