Theorem riota2f 6278

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 2608	. 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 22	. 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 6277	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 1446   F/wnf 1669   e. wcel 1889   F/_wnfc 2581   E!wreu 2741   iota_crio 6256

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ral 2744  df-rex 2745  df-reu 2746  df-v 3049  df-sbc 3270  df-un 3411  df-sn 3971  df-pr 3973  df-uni 4202  df-iota 5549  df-riota 6257

This theorem is referenced by:  riota2  6279  riotaprop  6280  riotass2  6283  riotass  6284  riotaxfrd  6287  cdlemksv2  34426  cdlemkuv2  34446  cdlemk36  34492