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Theorem rabsnel 29920
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 4430 . . 3 𝐵 ∈ {𝐵}
3 eleq2 2848 . . 3 ({𝑥𝐴𝜑} = {𝐵} → (𝐵 ∈ {𝑥𝐴𝜑} ↔ 𝐵 ∈ {𝐵}))
42, 3mpbiri 250 . 2 ({𝑥𝐴𝜑} = {𝐵} → 𝐵 ∈ {𝑥𝐴𝜑})
5 elrabi 3567 . 2 (𝐵 ∈ {𝑥𝐴𝜑} → 𝐵𝐴)
64, 5syl 17 1 ({𝑥𝐴𝜑} = {𝐵} → 𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1601  wcel 2107  {crab 3094  Vcvv 3398  {csn 4398
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-rab 3099  df-v 3400  df-sn 4399
This theorem is referenced by:  ddemeas  30905
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