Theorem riota2 5510

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 2219	. 2 ⊢ Ⅎ𝑥𝐵
2		nfv 1461	. 2 ⊢ Ⅎ𝑥𝜓
3		riota2.1	. 2 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))
4	1, 2, 3	riota2f 5509	1 ⊢ ((𝐵 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → (𝜓 ↔ (℩𝑥 ∈ 𝐴 𝜑) = 𝐵))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1284   ∈ wcel 1433  ∃!wreu 2350  ℩crio 5487

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-reu 2355  df-v 2603  df-sbc 2816  df-un 2977  df-sn 3404  df-pr 3405  df-uni 3602  df-iota 4887  df-riota 5488

This theorem is referenced by:  eqsupti  6409  prsrriota  6964  recriota  7056  axcaucvglemval  7063  subadd  7311  divmulap  7763  flqlelt  9278  flqbi  9292  remim  9747  resqrtcl  9915  rersqrtthlem  9916  divalgmod  10327  dfgcd3  10399  bezout  10400  oddpwdclemxy  10547