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Theorem rabsnt 3491
Description: Truth implied by equality of a restricted class abstraction and a singleton. (Contributed by NM, 29-May-2006.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
Hypotheses
Ref Expression
rabsnt.1 𝐵 ∈ V
rabsnt.2 (𝑥 = 𝐵 → (𝜑𝜓))
Assertion
Ref Expression
rabsnt ({𝑥𝐴𝜑} = {𝐵} → 𝜓)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜓,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rabsnt
StepHypRef Expression
1 rabsnt.1 . . . 4 𝐵 ∈ V
21snid 3449 . . 3 𝐵 ∈ {𝐵}
3 id 19 . . 3 ({𝑥𝐴𝜑} = {𝐵} → {𝑥𝐴𝜑} = {𝐵})
42, 3syl5eleqr 2172 . 2 ({𝑥𝐴𝜑} = {𝐵} → 𝐵 ∈ {𝑥𝐴𝜑})
5 rabsnt.2 . . . 4 (𝑥 = 𝐵 → (𝜑𝜓))
65elrab 2759 . . 3 (𝐵 ∈ {𝑥𝐴𝜑} ↔ (𝐵𝐴𝜓))
76simprbi 269 . 2 (𝐵 ∈ {𝑥𝐴𝜑} → 𝜓)
84, 7syl 14 1 ({𝑥𝐴𝜑} = {𝐵} → 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1285  wcel 1434  {crab 2357  Vcvv 2612  {csn 3422
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rab 2362  df-v 2614  df-sn 3428
This theorem is referenced by:  ontr2exmid  4304  onsucsssucexmid  4306  ordsoexmid  4341  unfiexmid  6555
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