Theorem reiota2f 5109

Description: A condition that allows us to represent "the unique element in A such that ph " with a class expression B. The second hypothesis is a weakened bound variable condition that allows hbsbc1g 2461 to be used. Compare reuuni2f 3810.

Hypotheses

Ref	Expression
reiota2f.1	|- (y e. B -> A.x y e. B)
reiota2f.2	|- (B e. A -> (ps -> A.xps))
reiota2f.3	|- (x = B -> (ph <-> ps))

Assertion

Ref	Expression
reiota2f	|- ((B e. A /\ E!x e. A ph) -> (ps <-> (iotax(x e. A /\ ph)) = B))

Distinct variable groups:   ph,y   x,y,A   y,B

Proof of Theorem reiota2f

Step	Hyp	Ref	Expression
1		reiota2f.1	. . . 4 |- (y e. B -> A.x y e. B)
2		ax-17 1317	. . . . . 6 |- (y e. A -> A.x y e. A)
3	1, 2	hbel 1996	. . . . 5 |- (B e. A -> A.x B e. A)
4		hbreu1 2252	. . . . . . 7 |- (E!x e. A ph -> A.xE!x e. A ph)
5	4	a1i 8	. . . . . 6 |- (B e. A -> (E!x e. A ph -> A.xE!x e. A ph))
6		reiota2f.2	. . . . . . 7 |- (B e. A -> (ps -> A.xps))
7		hbiota1 5091	. . . . . . . . 9 |- (y e. (iotax(x e. A /\ ph)) -> A.x y e. (iotax(x e. A /\ ph)))
8	7, 1	hbeq 1995	. . . . . . . 8 |- ((iotax(x e. A /\ ph)) = B -> A.x(iotax(x e. A /\ ph)) = B)
9	8	a1i 8	. . . . . . 7 |- (B e. A -> ((iotax(x e. A /\ ph)) = B -> A.x(iotax(x e. A /\ ph)) = B))
10	3, 6, 9	hbbid 1470	. . . . . 6 |- (B e. A -> ((ps <-> (iotax(x e. A /\ ph)) = B) -> A.x(ps <-> (iotax(x e. A /\ ph)) = B)))
11	3, 5, 10	hbimd 1468	. . . . 5 |- (B e. A -> ((E!x e. A ph -> (ps <-> (iotax(x e. A /\ ph)) = B)) -> A.x(E!x e. A ph -> (ps <-> (iotax(x e. A /\ ph)) = B))))
12	3, 11	hbim1 1458	. . . 4 |- ((B e. A -> (E!x e. A ph -> (ps <-> (iotax(x e. A /\ ph)) = B))) -> A.x(B e. A -> (E!x e. A ph -> (ps <-> (iotax(x e. A /\ ph)) = B))))
13		eleq1 1957	. . . . 5 |- (x = B -> (x e. A <-> B e. A))
14		reiota2f.3	. . . . . . 7 |- (x = B -> (ph <-> ps))
15		eqeq2 1893	. . . . . . 7 |- (x = B -> ((iotax(x e. A /\ ph)) = x <-> (iotax(x e. A /\ ph)) = B))
16	14, 15	bibi12d 691	. . . . . 6 |- (x = B -> ((ph <-> (iotax(x e. A /\ ph)) = x) <-> (ps <-> (iotax(x e. A /\ ph)) = B)))
17	16	imbi2d 674	. . . . 5 |- (x = B -> ((E!x e. A ph -> (ph <-> (iotax(x e. A /\ ph)) = x)) <-> (E!x e. A ph -> (ps <-> (iotax(x e. A /\ ph)) = B))))
18	13, 17	imbi12d 688	. . . 4 |- (x = B -> ((x e. A -> (E!x e. A ph -> (ph <-> (iotax(x e. A /\ ph)) = x))) <-> (B e. A -> (E!x e. A ph -> (ps <-> (iotax(x e. A /\ ph)) = B)))))
19		reiota1 5108	. . . . 5 |- ((x e. A /\ E!x e. A ph) -> (ph <-> (iotax(x e. A /\ ph)) = x))
20	19	ex 402	. . . 4 |- (x e. A -> (E!x e. A ph -> (ph <-> (iotax(x e. A /\ ph)) = x)))
21	1, 12, 18, 20	vtoclgf 2345	. . 3 |- (B e. A -> (B e. A -> (E!x e. A ph -> (ps <-> (iotax(x e. A /\ ph)) = B))))
22	21	pm2.43i 78	. 2 |- (B e. A -> (E!x e. A ph -> (ps <-> (iotax(x e. A /\ ph)) = B)))
23	22	imp 377	1 |- ((B e. A /\ E!x e. A ph) -> (ps <-> (iotax(x e. A /\ ph)) = B))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E!wreu 2107  iotacio 5087

This theorem is referenced by:  reiota2 5110  reiotass2 5111  riota2f 5579

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-reu 2111  df-v 2294  df-sbc 2454  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-uni 3178  df-iota 5089

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