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Theorem rabsnt 4627
 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 4561 . . 3 𝐵 ∈ {𝐵}
3 id 22 . . 3 ({𝑥𝐴𝜑} = {𝐵} → {𝑥𝐴𝜑} = {𝐵})
42, 3eleqtrrid 2859 . 2 ({𝑥𝐴𝜑} = {𝐵} → 𝐵 ∈ {𝑥𝐴𝜑})
5 rabsnt.2 . . . 4 (𝑥 = 𝐵 → (𝜑𝜓))
65elrab 3604 . . 3 (𝐵 ∈ {𝑥𝐴𝜑} ↔ (𝐵𝐴𝜓))
76simprbi 500 . 2 (𝐵 ∈ {𝑥𝐴𝜑} → 𝜓)
84, 7syl 17 1 ({𝑥𝐴𝜑} = {𝐵} → 𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2111  {crab 3074  Vcvv 3409  {csn 4525 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-rab 3079  df-v 3411  df-sn 4526 This theorem is referenced by:  ddemeas  31727
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