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Theorem rabsnel 30268
Description: Truth implied by equality of a restricted class abstraction and a singleton. (Contributed by Thierry Arnoux, 15-Sep-2018.)
Hypothesis
Ref Expression
rabsnel.1 𝐵 ∈ V
Assertion
Ref Expression
rabsnel ({𝑥𝐴𝜑} = {𝐵} → 𝐵𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rabsnel
StepHypRef Expression
1 rabsnel.1 . . . 4 𝐵 ∈ V
21snid 4575 . . 3 𝐵 ∈ {𝐵}
3 eleq2 2902 . . 3 ({𝑥𝐴𝜑} = {𝐵} → (𝐵 ∈ {𝑥𝐴𝜑} ↔ 𝐵 ∈ {𝐵}))
42, 3mpbiri 261 . 2 ({𝑥𝐴𝜑} = {𝐵} → 𝐵 ∈ {𝑥𝐴𝜑})
5 elrabi 3650 . 2 (𝐵 ∈ {𝑥𝐴𝜑} → 𝐵𝐴)
64, 5syl 17 1 ({𝑥𝐴𝜑} = {𝐵} → 𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2114  {crab 3134  Vcvv 3469  {csn 4539
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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794
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 2801  df-cleq 2815  df-clel 2894  df-rab 3139  df-v 3471  df-sn 4540
This theorem is referenced by:  ddemeas  31569
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