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Theorem rabsnel 29669
 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 4353 . . 3 𝐵 ∈ {𝐵}
3 eleq2 2828 . . 3 ({𝑥𝐴𝜑} = {𝐵} → (𝐵 ∈ {𝑥𝐴𝜑} ↔ 𝐵 ∈ {𝐵}))
42, 3mpbiri 248 . 2 ({𝑥𝐴𝜑} = {𝐵} → 𝐵 ∈ {𝑥𝐴𝜑})
5 elrabi 3499 . 2 (𝐵 ∈ {𝑥𝐴𝜑} → 𝐵𝐴)
64, 5syl 17 1 ({𝑥𝐴𝜑} = {𝐵} → 𝐵𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1632   ∈ wcel 2139  {crab 3054  Vcvv 3340  {csn 4321 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rab 3059  df-v 3342  df-sn 4322 This theorem is referenced by:  ddemeas  30629
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