Theorem riota2 5990

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 2372	. 2 |- F/_ x B
2		nfv 1574	. 2 |- F/ x ps
3		riota2.1	. 2 |- ( x = B -> ( ph <-> ps ) )
4	1, 2, 3	riota2f 5989	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 1395   e. wcel 2200   E!wreu 2510   iota_crio 5965

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-reu 2515  df-v 2802  df-sbc 3030  df-un 3202  df-sn 3673  df-pr 3674  df-uni 3892  df-iota 5284  df-riota 5966

This theorem is referenced by:  eqsupti  7186  prsrriota  7998  recriota  8100  axcaucvglemval  8107  subadd  8372  divmulap  8845  flqlelt  10526  flqbi  10540  remim  11411  resqrtcl  11580  rersqrtthlem  11581  divalgmod  12478  dfgcd3  12571  bezout  12572  oddpwdclemxy  12731  qnumdenbi  12754  ismgmid  13450  isgrpinv  13627  usgredg2vlem2  16062