Theorem riota2 5940

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 2349	. 2 |- F/_ x B
2		nfv 1552	. 2 |- F/ x ps
3		riota2.1	. 2 |- ( x = B -> ( ph <-> ps ) )
4	1, 2, 3	riota2f 5939	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 1373   e. wcel 2177   E!wreu 2487   iota_crio 5916

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-rex 2491  df-reu 2492  df-v 2775  df-sbc 3003  df-un 3174  df-sn 3644  df-pr 3645  df-uni 3860  df-iota 5246  df-riota 5917

This theorem is referenced by:  eqsupti  7119  prsrriota  7931  recriota  8033  axcaucvglemval  8040  subadd  8305  divmulap  8778  flqlelt  10451  flqbi  10465  remim  11256  resqrtcl  11425  rersqrtthlem  11426  divalgmod  12323  dfgcd3  12416  bezout  12417  oddpwdclemxy  12576  qnumdenbi  12599  ismgmid  13294  isgrpinv  13471