Theorem riota2 6005

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 2375	. 2 ⊢ Ⅎ𝑥𝐵
2		nfv 1577	. 2 ⊢ Ⅎ𝑥𝜓
3		riota2.1	. 2 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))
4	1, 2, 3	riota2f 6004	1 ⊢ ((𝐵 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → (𝜓 ↔ (℩𝑥 ∈ 𝐴 𝜑) = 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2202  ∃!wreu 2513  ℩crio 5980

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rex 2517  df-reu 2518  df-v 2805  df-sbc 3033  df-un 3205  df-sn 3679  df-pr 3680  df-uni 3899  df-iota 5293  df-riota 5981

This theorem is referenced by:  eqsupti  7255  prsrriota  8068  recriota  8170  axcaucvglemval  8177  subadd  8441  divmulap  8914  flqlelt  10599  flqbi  10613  remim  11500  resqrtcl  11669  rersqrtthlem  11670  divalgmod  12568  dfgcd3  12661  bezout  12662  oddpwdclemxy  12821  qnumdenbi  12844  ismgmid  13540  isgrpinv  13717  usgredg2vlem2  16164