Theorem reuss2 2271

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 555	. . . . . . . . . . . . . 14 |- (((x e. A -> x e. B) /\ (ph -> ps)) -> ((x e. A /\ ph) -> (x e. B /\ ps)))
2		ssel 2059	. . . . . . . . . . . . . 14 |- (A (_ B -> (x e. A -> x e. B))
3	1, 2	sylan 448	. . . . . . . . . . . . 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 364	. . . . . . . . 9 |- (A (_ B -> ((x e. A -> (ph -> ps)) -> ((x e. A /\ ph) -> (x e. B /\ ps))))
8	7	19.20dv 1287	. . . . . . . 8 |- (A (_ B -> (A.x(x e. A -> (ph -> ps)) -> A.x((x e. A /\ ph) -> (x e. B /\ ps))))
9	8	imp 350	. . . . . . 7 |- ((A (_ B /\ A.x(x e. A -> (ph -> ps))) -> A.x((x e. A /\ ph) -> (x e. B /\ ps)))
10		df-ral 1646	. . . . . . 7 |- (A.x e. A (ph -> ps) <-> A.x(x e. A -> (ph -> ps)))
11	9, 10	sylan2b 452	. . . . . 6 |- ((A (_ B /\ A.x e. A (ph -> ps)) -> A.x((x e. A /\ ph) -> (x e. B /\ ps)))
12		euimmo 1418	. . . . . 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 1407	. . . . . . 7 |- (E!x(x e. A /\ ph) <-> (E.x(x e. A /\ ph) /\ E*x(x e. A /\ ph)))
15	14	biimpr 152	. . . . . 6 |- ((E.x(x e. A /\ ph) /\ E*x(x e. A /\ ph)) -> E!x(x e. A /\ ph))
16	15	ex 373	. . . . 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 363	. . 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 1648	. . 3 |- (E!x e. A ph <-> E!x(x e. A /\ ph))
20	18, 19	sylibr 200	. 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 1647	. . 3 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
22		df-reu 1648	. . 3 |- (E!x e. B ps <-> E!x(x e. B /\ ps))
23	21, 22	anbi12i 482	. 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 452	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 223  A.wal 952   e. wcel 956  E.wex 978  E!weu 1378  E*wmo 1379  A.wral 1642  E.wrex 1643  E!wreu 1644   (_ wss 2043

This theorem is referenced by:  reuss 2272  reuun1 2273  reuuniss2 2886  grpidinv2 8010  grpinv 8019

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ral 1646  df-rex 1647  df-reu 1648  df-in 2047  df-ss 2049

Copyright terms: Public domain