Theorem riota2f 6286

Description: This theorem shows a condition that allows us to represent a descriptor with a class expression B. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
riota2f.1	|- F/_ x B
riota2f.2	|- F/ x ps
riota2f.3	|- ( x = B -> ( ph <-> ps ) )

Assertion

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

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   ps( x)   B( x)

Proof of Theorem riota2f

Step	Hyp	Ref	Expression
1		riota2f.1	. . 3 |- F/_ x B
2	1	nfel1 2601	. 2 |- F/ x B e. A
3	1	a1i 11	. 2 |- ( B e. A -> F/_ x B )
4		riota2f.2	. . 3 |- F/ x ps
5	4	a1i 11	. 2 |- ( B e. A -> F/ x ps )
6		id 23	. 2 |- ( B e. A -> B e. A )
7		riota2f.3	. . 3 |- ( x = B -> ( ph <-> ps ) )
8	7	adantl 468	. 2 |- ( ( B e. A /\ x = B ) -> ( ph <-> ps ) )
9	2, 3, 5, 6, 8	riota2df 6285	1 |- ( ( B e. A /\ E! x e. A ph ) -> ( ps <-> ( iota_ x e. A ph ) = B ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   = wceq 1438   F/wnf 1664   e. wcel 1869   F/_wnfc 2571   E!wreu 2778   iota_crio 6264

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ral 2781  df-rex 2782  df-reu 2783  df-v 3084  df-sbc 3301  df-un 3442  df-sn 3998  df-pr 4000  df-uni 4218  df-iota 5563  df-riota 6265

This theorem is referenced by:  riota2  6287  riotaprop  6288  riotass2  6291  riotass  6292  riotaxfrd  6295  cdlemksv2  34377  cdlemkuv2  34397  cdlemk36  34443