Theorem riota2 7393

Description: This theorem shows a condition that allows 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 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
riota2	⊢ ((𝐵 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → (𝜓 ↔ (℩𝑥 ∈ 𝐴 𝜑) = 𝐵))

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

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem riota2

Step	Hyp	Ref	Expression
1		nfcv 2903	. 2 ⊢ Ⅎ𝑥𝐵
2		nfv 1917	. 2 ⊢ Ⅎ𝑥𝜓
3		riota2.1	. 2 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))
4	1, 2, 3	riota2f 7392	1 ⊢ ((𝐵 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → (𝜓 ↔ (℩𝑥 ∈ 𝐴 𝜑) = 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∃!wreu 3374  ℩crio 7366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-reu 3377  df-v 3476  df-un 3953  df-in 3955  df-ss 3965  df-sn 4629  df-pr 4631  df-uni 4909  df-iota 6495  df-riota 7367

This theorem is referenced by:  eqsup  9453  sup0  9463  ttrcltr  9713  fin23lem22  10324  subadd  11467  divmul  11879  fllelt  13766  flflp1  13776  flval2  13783  flbi  13785  remim  15068  resqrtcl  15204  resqrtthlem  15205  sqrtneg  15218  sqrtthlem  15313  divalgmod  16353  qnumdenbi  16684  catidd  17628  lubprop  18315  glbprop  18328  poslubd  18370  isglbd  18466  ismgmid  18590  isgrpinv  18914  pj1id  19608  coeeq  25965  scutbday  27530  eqscut  27531  scutun12  27536  scutbdaylt  27544  divsmulw  27867  ismir  28165  mireq  28171  ismidb  28284  islmib  28293  usgredg2vlem2  28738  frgrncvvdeqlem3  29809  frgr2wwlkeqm  29839  cnidOLD  30090  hilid  30669  pjpreeq  30906  cnvbraval  31618  cdj3lem2  31943  xdivmul  32346  cvmliftphtlem  34594  cvmlift3lem4  34599  cvmlift3lem6  34601  cvmlift3lem9  34604  transportprops  35298  ltflcei  36779  cmpidelt  37030  exidresid  37050  lshpkrlem1  38283  cdlemeiota  39759  dochfl1  40650  hgmapvs  41065  evlsval3  41433  fsuppind  41464  renegadd  41547  resubadd  41554  addinvcom  41606  wessf1ornlem  44183  fourierdlem50  45171