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Theorem rabsneu 4421
Description: Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.) (Revised by Mario Carneiro, 23-Dec-2016.)
Assertion
Ref Expression
rabsneu ((𝐴𝑉 ∧ {𝑥𝐵𝜑} = {𝐴}) → ∃!𝑥𝐵 𝜑)

Proof of Theorem rabsneu
StepHypRef Expression
1 df-rab 3064 . . . 4 {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)}
21eqeq1i 2770 . . 3 ({𝑥𝐵𝜑} = {𝐴} ↔ {𝑥 ∣ (𝑥𝐵𝜑)} = {𝐴})
3 absneu 4420 . . 3 ((𝐴𝑉 ∧ {𝑥 ∣ (𝑥𝐵𝜑)} = {𝐴}) → ∃!𝑥(𝑥𝐵𝜑))
42, 3sylan2b 587 . 2 ((𝐴𝑉 ∧ {𝑥𝐵𝜑} = {𝐴}) → ∃!𝑥(𝑥𝐵𝜑))
5 df-reu 3062 . 2 (∃!𝑥𝐵 𝜑 ↔ ∃!𝑥(𝑥𝐵𝜑))
64, 5sylibr 225 1 ((𝐴𝑉 ∧ {𝑥𝐵𝜑} = {𝐴}) → ∃!𝑥𝐵 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wcel 2155  ∃!weu 2581  {cab 2751  ∃!wreu 3057  {crab 3059  {csn 4336
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 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
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 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-reu 3062  df-rab 3064  df-v 3352  df-sn 4337
This theorem is referenced by: (None)
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