Theorem cp 3547

Description: Collection Principle. This remarkable theorem scheme is in effect a very strong generalization of the Axiom of Replacement. The proof makes use of Scott's trick scottex 3541 that collapses a proper class into a set of minimum rank. The wff ph can be thought of as ph(x, y). Scheme "Collection Principle" of [Jech] p. 72.

Assertion

Ref	Expression
cp	|- E.wA.x e. z (E.yph -> E.y e. w ph)

Distinct variable group(s):   ph,z,w   x,y,z,w

Proof of Theorem cp

Step	Hyp	Ref	Expression
1		visset 1350	. . 3 |- z e. V
2	1	cplem2 3546	. 2 |- E.wA.x e. z (-. {y | ph} = (/) -> -. ({y | ph} i^i w) = (/))
3		abn0 1715	. . . . 5 |- (-. {y | ph} = (/) <-> E.yph)
4		elin 1635	. . . . . . . 8 |- (y e. ({y | ph} i^i w) <-> (y e. {y | ph} /\ y e. w))
5		abid 1094	. . . . . . . . 9 |- (y e. {y | ph} <-> ph)
6	5	anbi1i 368	. . . . . . . 8 |- ((y e. {y | ph} /\ y e. w) <-> (ph /\ y e. w))
7		ancom 333	. . . . . . . 8 |- ((ph /\ y e. w) <-> (y e. w /\ ph))
8	4, 6, 7	3bitr 155	. . . . . . 7 |- (y e. ({y | ph} i^i w) <-> (y e. w /\ ph))
9	8	biex 733	. . . . . 6 |- (E.y y e. ({y | ph} i^i w) <-> E.y(y e. w /\ ph))
10		hbab1 1095	. . . . . . . 8 |- (z e. {y | ph} -> A.y z e. {y | ph})
11		ax-17 925	. . . . . . . 8 |- (z e. w -> A.y z e. w)
12	10, 11	hbin 1647	. . . . . . 7 |- (z e. ({y | ph} i^i w) -> A.y z e. ({y | ph} i^i w))
13	12	n0f 1713	. . . . . 6 |- (-. ({y | ph} i^i w) = (/) <-> E.y y e. ({y | ph} i^i w))
14		df-rex 1206	. . . . . 6 |- (E.y e. w ph <-> E.y(y e. w /\ ph))
15	9, 13, 14	3bitr4 158	. . . . 5 |- (-. ({y | ph} i^i w) = (/) <-> E.y e. w ph)
16	3, 15	imbi12i 163	. . . 4 |- ((-. {y | ph} = (/) -> -. ({y | ph} i^i w) = (/)) <-> (E.yph -> E.y e. w ph))
17	16	biral 1223	. . 3 |- (A.x e. z (-. {y | ph} = (/) -> -. ({y | ph} i^i w) = (/)) <-> A.x e. z (E.yph -> E.y e. w ph))
18	17	biex 733	. 2 |- (E.wA.x e. z (-. {y | ph} = (/) -> -. ({y | ph} i^i w) = (/)) <-> E.wA.x e. z (E.yph -> E.y e. w ph))
19	2, 18	mpbi 164	1 |- E.wA.x e. z (E.yph -> E.y e. w ph)

Syntax hints:  -. wn 1   -> wi 2   /\ wa 196  E.wex 678   e. wel 803  {cab 1090   = wceq 1091   e. wcel 1092  A.wral 1201  E.wrex 1202   i^i cin 1486  (/)c0 1707

This theorem is referenced by:  bnd 3548

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-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-iin 1997  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-r1 3487  df-rank 3488

metamath.org