Theorem riota2 5542

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 2223	. 2 |- F/_ x B
2		nfv 1462	. 2 |- F/ x ps
3		riota2.1	. 2 |- ( x = B -> ( ph <-> ps ) )
4	1, 2, 3	riota2f 5541	1 |- ( ( B e. A /\ E! x e. A ph ) -> ( ps <-> ( iota_ x e. A ph ) = B ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1285   e. wcel 1434   E!wreu 2355   iota_crio 5519

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rex 2359  df-reu 2360  df-v 2612  df-sbc 2825  df-un 2986  df-sn 3422  df-pr 3423  df-uni 3622  df-iota 4917  df-riota 5520

This theorem is referenced by:  eqsupti  6504  prsrriota  7096  recriota  7188  axcaucvglemval  7195  subadd  7448  divmulap  7900  flqlelt  9428  flqbi  9442  remim  9966  resqrtcl  10134  rersqrtthlem  10135  divalgmod  10552  dfgcd3  10624  bezout  10625  oddpwdclemxy  10772  qnumdenbi  10795