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Theorem rabsnt 4398
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 4341 . . 3 𝐵 ∈ {𝐵}
3 id 22 . . 3 ({𝑥𝐴𝜑} = {𝐵} → {𝑥𝐴𝜑} = {𝐵})
42, 3syl5eleqr 2834 . 2 ({𝑥𝐴𝜑} = {𝐵} → 𝐵 ∈ {𝑥𝐴𝜑})
5 rabsnt.2 . . . 4 (𝑥 = 𝐵 → (𝜑𝜓))
65elrab 3492 . . 3 (𝐵 ∈ {𝑥𝐴𝜑} ↔ (𝐵𝐴𝜓))
76simprbi 483 . 2 (𝐵 ∈ {𝑥𝐴𝜑} → 𝜓)
84, 7syl 17 1 ({𝑥𝐴𝜑} = {𝐵} → 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1620  wcel 2127  {crab 3042  Vcvv 3328  {csn 4309
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-rab 3047  df-v 3330  df-sn 4310
This theorem is referenced by:  ddemeas  30579
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