Theorem riota2 6062

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 2392	. 2 |- F/_ x B
2		nfv 1581	. 2 |- F/ x ps
3		riota2.1	. 2 |- ( x = B -> ( ph <-> ps ) )
4	1, 2, 3	riota2f 6061	1 |- ( ( B e. A /\ E! x e. A ph ) -> ( ps <-> ( iota_ x e. A ph ) = B ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   E!wreu 2530   iota_crio 6037

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-reu 2535  df-v 2823  df-sbc 3052  df-un 3224  df-sn 3715  df-pr 3716  df-uni 3936  df-iota 5337  df-riota 6038

This theorem is used by:  eqsupti  7337  prsrriota  8156  recriota  8258  axcaucvglemval  8265  subadd  8531  divmulap  9008  flqlelt  10723  flaplelt  10724  flqbi  10739  remim  11640  resqrtcl  11810  rersqrtthlem  11811  divalgmod  12712  dfgcd3  12805  bezout  12806  nnmaxpwlemxy  12966  qnumdenbi  12990  ismgmid  13748  isgrpinv  13910  usgredg2vlem2  16586