Theorem riota2 6796

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 → (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 2902	. 2 ⊢ Ⅎ𝑥𝐵
2		nfv 1992	. 2 ⊢ Ⅎ𝑥𝜓
3		riota2.1	. 2 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))
4	1, 2, 3	riota2f 6795	1 ⊢ ((𝐵 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → (𝜓 ↔ (℩𝑥 ∈ 𝐴 𝜑) = 𝐵))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1632   ∈ wcel 2139  ∃!wreu 3052  ℩crio 6773

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-reu 3057  df-v 3342  df-sbc 3577  df-un 3720  df-sn 4322  df-pr 4324  df-uni 4589  df-iota 6012  df-riota 6774

This theorem is referenced by:  eqsup  8527  sup0  8537  fin23lem22  9341  subadd  10476  divmul  10880  fllelt  12792  flflp1  12802  flval2  12809  flbi  12811  remim  14056  resqrtcl  14193  resqrtthlem  14194  sqrtneg  14207  sqrtthlem  14301  divalgmod  15331  divalgmodOLD  15332  qnumdenbi  15654  catidd  16542  lubprop  17187  glbprop  17200  isglbd  17318  poslubd  17349  ismgmid  17465  isgrpinv  17673  pj1id  18312  coeeq  24182  ismir  25753  mireq  25759  ismidb  25869  islmib  25878  usgredg2vlem2  26317  frgrncvvdeqlem3  27455  frgr2wwlkeqm  27485  cnidOLD  27746  hilid  28327  pjpreeq  28566  cnvbraval  29278  cdj3lem2  29603  xdivmul  29942  cvmliftphtlem  31606  cvmlift3lem4  31611  cvmlift3lem6  31613  cvmlift3lem9  31616  scutbday  32219  scutun12  32223  scutbdaylt  32228  transportprops  32447  ltflcei  33710  cmpidelt  33971  exidresid  33991  lshpkrlem1  34900  cdlemeiota  36375  dochfl1  37267  hgmapvs  37685  wessf1ornlem  39870  fourierdlem50  40876