Theorem riota2f 5579

Description: This theorem shows a condition that allows us to represent a descriptor 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
riota2f.1	|- (y e. B -> A.x y e. B)
riota2f.2	|- (B e. A -> (ps -> A.xps))
riota2f.3	|- (x = B -> (ph <-> ps))

Assertion

Ref	Expression
riota2f	|- ((B e. A /\ E!x e. A ph) -> (ps <-> (iota_x e. Aph) = B))

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

Proof of Theorem riota2f

Step	Hyp	Ref	Expression
1		riota2f.1	. . 3 |- (y e. B -> A.x y e. B)
2		riota2f.2	. . 3 |- (B e. A -> (ps -> A.xps))
3		riota2f.3	. . 3 |- (x = B -> (ph <-> ps))
4	1, 2, 3	reiota2f 5109	. 2 |- ((B e. A /\ E!x e. A ph) -> (ps <-> (iotax(x e. A /\ ph)) = B))
5		riotaiota 5566	. . . 4 |- (E!x e. A ph -> (iota_x e. Aph) = (iotax(x e. A /\ ph)))
6	5	eqeq1d 1892	. . 3 |- (E!x e. A ph -> ((iota_x e. Aph) = B <-> (iotax(x e. A /\ ph)) = B))
7	6	adantl 424	. 2 |- ((B e. A /\ E!x e. A ph) -> ((iota_x e. Aph) = B <-> (iotax(x e. A /\ ph)) = B))
8	4, 7	bitr4d 590	1 |- ((B e. A /\ E!x e. A ph) -> (ps <-> (iota_x e. Aph) = B))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E!wreu 2107  iotacio 5087  iota_crio 5555

This theorem is referenced by:  riota5 5580  riotaxfrd 5581  subaddi 6528  divmuli 6894  lubprop 16805  glbprop 16810

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-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-uni 3178  df-iota 5089  df-riota 5560

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