Theorem riota2 6029

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, 10-Dec-2016.)

Hypothesis

Ref	Expression
riota2.1	|- ( x = B -> ( ph <-> ps ) )

Assertion

Ref	Expression
riota2	|- ( ( B e. A /\ E! x e. A ph ) -> ( ps <-> ( iota_ x e. A ph ) = B ) )

Distinct variable groups:   ps, x   x, A   x, B

Allowed substitution hint:   ph( x)

Proof of Theorem riota2

Step	Hyp	Ref	Expression
1		nfcv 2386	. 2 |- F/_ x B
2		nfv 1577	. 2 |- F/ x ps
3		riota2.1	. 2 |- ( x = B -> ( ph <-> ps ) )
4	1, 2, 3	riota2f 6028	1 |- ( ( B e. A /\ E! x e. A ph ) -> ( ps <-> ( iota_ x e. A ph ) = B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   E!wreu 2524   iota_crio 6004

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rex 2528  df-reu 2529  df-v 2817  df-sbc 3045  df-un 3217  df-sn 3697  df-pr 3698  df-uni 3917  df-iota 5314  df-riota 6005

This theorem is referenced by:  eqsupti  7289  prsrriota  8108  recriota  8210  axcaucvglemval  8217  subadd  8481  divmulap  8954  flqlelt  10643  flqbi  10657  remim  11553  resqrtcl  11722  rersqrtthlem  11723  divalgmod  12621  dfgcd3  12714  bezout  12715  oddpwdclemxy  12874  qnumdenbi  12897  ismgmid  13611  isgrpinv  13788  usgredg2vlem2  16267