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Theorem rabsnif 4394
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 4393 . . 3 {𝑥 ∈ {𝐴} ∣ 𝜑} = if([𝐴 / 𝑥]𝜑, {𝐴}, ∅)
2 rabsnif.f . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
32sbcieg 3620 . . . 4 (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑𝜓))
43ifbid 4247 . . 3 (𝐴 ∈ V → if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = if(𝜓, {𝐴}, ∅))
51, 4syl5eq 2817 . 2 (𝐴 ∈ V → {𝑥 ∈ {𝐴} ∣ 𝜑} = if(𝜓, {𝐴}, ∅))
6 rab0 4102 . . . 4 {𝑥 ∈ ∅ ∣ 𝜑} = ∅
7 ifid 4264 . . . 4 if(𝜓, ∅, ∅) = ∅
86, 7eqtr4i 2796 . . 3 {𝑥 ∈ ∅ ∣ 𝜑} = if(𝜓, ∅, ∅)
9 snprc 4389 . . . . 5 𝐴 ∈ V ↔ {𝐴} = ∅)
109biimpi 206 . . . 4 𝐴 ∈ V → {𝐴} = ∅)
1110rabeqdv 3344 . . 3 𝐴 ∈ V → {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝑥 ∈ ∅ ∣ 𝜑})
1210ifeq1d 4243 . . 3 𝐴 ∈ V → if(𝜓, {𝐴}, ∅) = if(𝜓, ∅, ∅))
138, 11, 123eqtr4a 2831 . 2 𝐴 ∈ V → {𝑥 ∈ {𝐴} ∣ 𝜑} = if(𝜓, {𝐴}, ∅))
145, 13pm2.61i 176 1 {𝑥 ∈ {𝐴} ∣ 𝜑} = if(𝜓, {𝐴}, ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196   = wceq 1631  wcel 2145  {crab 3065  Vcvv 3351  [wsbc 3587  c0 4063  ifcif 4225  {csn 4316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-nul 4064  df-if 4226  df-sn 4317
This theorem is referenced by:  suppsnop  7460  m1detdiag  20621  1loopgrvd2  26634  1hevtxdg1  26637  1egrvtxdg1  26640
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