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Theorem rabsnif 4658
Description: A restricted class abstraction restricted to a singleton is either the empty set or the singleton itself. (Contributed by AV, 12-Apr-2019.) (Proof shortened by AV, 21-Jul-2019.)
Hypothesis
Ref Expression
rabsnif.f (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
rabsnif {𝑥 ∈ {𝐴} ∣ 𝜑} = if(𝜓, {𝐴}, ∅)
Distinct variable groups:   𝑥,𝐴   𝜓,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rabsnif
StepHypRef Expression
1 rabsnifsb 4657 . . 3 {𝑥 ∈ {𝐴} ∣ 𝜑} = if([𝐴 / 𝑥]𝜑, {𝐴}, ∅)
2 rabsnif.f . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
32sbcieg 3814 . . . 4 (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑𝜓))
43ifbid 4492 . . 3 (𝐴 ∈ V → if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = if(𝜓, {𝐴}, ∅))
51, 4syl5eq 2873 . 2 (𝐴 ∈ V → {𝑥 ∈ {𝐴} ∣ 𝜑} = if(𝜓, {𝐴}, ∅))
6 rab0 4341 . . . 4 {𝑥 ∈ ∅ ∣ 𝜑} = ∅
7 ifid 4509 . . . 4 if(𝜓, ∅, ∅) = ∅
86, 7eqtr4i 2852 . . 3 {𝑥 ∈ ∅ ∣ 𝜑} = if(𝜓, ∅, ∅)
9 snprc 4652 . . . . 5 𝐴 ∈ V ↔ {𝐴} = ∅)
109biimpi 217 . . . 4 𝐴 ∈ V → {𝐴} = ∅)
1110rabeqdv 3490 . . 3 𝐴 ∈ V → {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝑥 ∈ ∅ ∣ 𝜑})
1210ifeq1d 4488 . . 3 𝐴 ∈ V → if(𝜓, {𝐴}, ∅) = if(𝜓, ∅, ∅))
138, 11, 123eqtr4a 2887 . 2 𝐴 ∈ V → {𝑥 ∈ {𝐴} ∣ 𝜑} = if(𝜓, {𝐴}, ∅))
145, 13pm2.61i 183 1 {𝑥 ∈ {𝐴} ∣ 𝜑} = if(𝜓, {𝐴}, ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207   = wceq 1530  wcel 2107  {crab 3147  Vcvv 3500  [wsbc 3776  c0 4295  ifcif 4470  {csn 4564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-nul 4296  df-if 4471  df-sn 4565
This theorem is referenced by:  suppsnop  7835  m1detdiag  21122  1loopgrvd2  27199  1hevtxdg1  27202  1egrvtxdg1  27205
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