Theorem reuss 1577

Description: Restriction of uniqueness to a smaller subclass.

Assertion

Ref	Expression
reuss	|- ((A (_ B /\ (E.x e. A ph /\ E!x e. B ph)) -> E!x e. A ph)

Distinct variable group(s):   x,A   x,B

Proof of Theorem reuss

Step	Hyp	Ref	Expression
1		ssel 1502	. . . . . . . 8 |- (A (_ B -> (x e. A -> x e. B))
2	1	anim1d 432	. . . . . . 7 |- (A (_ B -> ((x e. A /\ ph) -> (x e. B /\ ph)))
3	2	19.21aiv 943	. . . . . 6 |- (A (_ B -> A.x((x e. A /\ ph) -> (x e. B /\ ph)))
4		euimmo 1045	. . . . . 6 |- (A.x((x e. A /\ ph) -> (x e. B /\ ph)) -> (E!x(x e. B /\ ph) -> E*x(x e. A /\ ph)))
5	3, 4	syl 12	. . . . 5 |- (A (_ B -> (E!x(x e. B /\ ph) -> E*x(x e. A /\ ph)))
6		eu5 1035	. . . . . . 7 |- (E!x(x e. A /\ ph) <-> (E.x(x e. A /\ ph) /\ E*x(x e. A /\ ph)))
7	6	biimpr 134	. . . . . 6 |- ((E.x(x e. A /\ ph) /\ E*x(x e. A /\ ph)) -> E!x(x e. A /\ ph))
8	7	exp 291	. . . . 5 |- (E.x(x e. A /\ ph) -> (E*x(x e. A /\ ph) -> E!x(x e. A /\ ph)))
9	5, 8	syl9 55	. . . 4 |- (A (_ B -> (E.x(x e. A /\ ph) -> (E!x(x e. B /\ ph) -> E!x(x e. A /\ ph))))
10	9	imp32 281	. . 3 |- ((A (_ B /\ (E.x(x e. A /\ ph) /\ E!x(x e. B /\ ph))) -> E!x(x e. A /\ ph))
11		df-reu 1207	. . 3 |- (E!x e. A ph <-> E!x(x e. A /\ ph))
12	10, 11	sylibr 175	. 2 |- ((A (_ B /\ (E.x(x e. A /\ ph) /\ E!x(x e. B /\ ph))) -> E!x e. A ph)
13		df-rex 1206	. . 3 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
14		df-reu 1207	. . 3 |- (E!x e. B ph <-> E!x(x e. B /\ ph))
15	13, 14	anbi12i 369	. 2 |- ((E.x e. A ph /\ E!x e. B ph) <-> (E.x(x e. A /\ ph) /\ E!x(x e. B /\ ph)))
16	12, 15	sylan2b 347	1 |- ((A (_ B /\ (E.x e. A ph /\ E!x e. B ph)) -> E!x e. A ph)

Syntax hints:   -> wi 2   /\ wa 196  A.wal 672  E.wex 678  E!weu 1007  E*wmo 1008   e. wcel 1092  E.wrex 1202  E!wreu 1203   (_ wss 1487

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-reu 1207  df-in 1491  df-ss 1492

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