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Theorem rabsnt 4420
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 4365 . . 3 𝐵 ∈ {𝐵}
3 id 22 . . 3 ({𝑥𝐴𝜑} = {𝐵} → {𝑥𝐴𝜑} = {𝐵})
42, 3syl5eleqr 2850 . 2 ({𝑥𝐴𝜑} = {𝐵} → 𝐵 ∈ {𝑥𝐴𝜑})
5 rabsnt.2 . . . 4 (𝑥 = 𝐵 → (𝜑𝜓))
65elrab 3518 . . 3 (𝐵 ∈ {𝑥𝐴𝜑} ↔ (𝐵𝐴𝜓))
76simprbi 490 . 2 (𝐵 ∈ {𝑥𝐴𝜑} → 𝜓)
84, 7syl 17 1 ({𝑥𝐴𝜑} = {𝐵} → 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197   = wceq 1652  wcel 2155  {crab 3058  Vcvv 3349  {csn 4333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2742
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-rab 3063  df-v 3351  df-sn 4334
This theorem is referenced by:  ddemeas  30680
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