Theorem riota2 5630

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 2228	. 2 ⊢ Ⅎ𝑥𝐵
2		nfv 1466	. 2 ⊢ Ⅎ𝑥𝜓
3		riota2.1	. 2 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))
4	1, 2, 3	riota2f 5629	1 ⊢ ((𝐵 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → (𝜓 ↔ (℩𝑥 ∈ 𝐴 𝜑) = 𝐵))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1289   ∈ wcel 1438  ∃!wreu 2361  ℩crio 5607

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-reu 2366  df-v 2621  df-sbc 2841  df-un 3003  df-sn 3452  df-pr 3453  df-uni 3654  df-iota 4980  df-riota 5608

This theorem is referenced by:  eqsupti  6691  prsrriota  7333  recriota  7425  axcaucvglemval  7432  subadd  7685  divmulap  8142  flqlelt  9683  flqbi  9697  remim  10294  resqrtcl  10462  rersqrtthlem  10463  divalgmod  11205  dfgcd3  11277  bezout  11278  oddpwdclemxy  11425  qnumdenbi  11448