Theorem reuss2 3536

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 2621	. . 3 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
2		df-reu 2622	. . 3 |- (E!x e. B ps <-> E!x(x e. B /\ ps))
3	1, 2	anbi12i 678	. 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 2620	. . . . . . 7 |- (A.x e. A (ph -> ps) <-> A.x(x e. A -> (ph -> ps)))
5		ssel 3268	. . . . . . . . . . . . . 14 |- (A C_ B -> (x e. A -> x e. B))
6		prth 554	. . . . . . . . . . . . . 14 |- (((x e. A -> x e. B) /\ (ph -> ps)) -> ((x e. A /\ ph) -> (x e. B /\ ps)))
7	5, 6	sylan 457	. . . . . . . . . . . . 13 |- ((A C_ B /\ (ph -> ps)) -> ((x e. A /\ ph) -> (x e. B /\ ps)))
8	7	exp4b 590	. . . . . . . . . . . 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 572	. . . . . . . . 9 |- (A C_ B -> ((x e. A -> (ph -> ps)) -> ((x e. A /\ ph) -> (x e. B /\ ps))))
12	11	alimdv 1621	. . . . . . . 8 |- (A C_ B -> (A.x(x e. A -> (ph -> ps)) -> A.x((x e. A /\ ph) -> (x e. B /\ ps))))
13	12	imp 418	. . . . . . 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 461	. . . . . 6 |- ((A C_ B /\ A.x e. A (ph -> ps)) -> A.x((x e. A /\ ph) -> (x e. B /\ ps)))
15		euimmo 2253	. . . . . 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 15	. . . . 5 |- ((A C_ B /\ A.x e. A (ph -> ps)) -> (E!x(x e. B /\ ps) -> E*x(x e. A /\ ph)))
17		eu5 2242	. . . . . 6 |- (E!x(x e. A /\ ph) <-> (E.x(x e. A /\ ph) /\ E*x(x e. A /\ ph)))
18	17	simplbi2 608	. . . . 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 422	. . 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 2622	. . 3 |- (E!x e. A ph <-> E!x(x e. A /\ ph))
22	20, 21	sylibr 203	. 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 461	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 358  A.wal 1540  E.wex 1541   e. wcel 1710  E!weu 2204  E*wmo 2205  A.wral 2615  E.wrex 2616  E!wreu 2617   C_ wss 3258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rex 2621  df-reu 2622  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260

This theorem is referenced by:  reuss  3537  reuun1  3538