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Theorem rabsnif 4689
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 4688 . . 3 {𝑥 ∈ {𝐴} ∣ 𝜑} = if([𝐴 / 𝑥]𝜑, {𝐴}, ∅)
2 rabsnif.f . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
32sbcieg 3784 . . . 4 (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑𝜓))
43ifbid 4514 . . 3 (𝐴 ∈ V → if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = if(𝜓, {𝐴}, ∅))
51, 4eqtrid 2789 . 2 (𝐴 ∈ V → {𝑥 ∈ {𝐴} ∣ 𝜑} = if(𝜓, {𝐴}, ∅))
6 rab0 4347 . . . 4 {𝑥 ∈ ∅ ∣ 𝜑} = ∅
7 ifid 4531 . . . 4 if(𝜓, ∅, ∅) = ∅
86, 7eqtr4i 2768 . . 3 {𝑥 ∈ ∅ ∣ 𝜑} = if(𝜓, ∅, ∅)
9 snprc 4683 . . . . 5 𝐴 ∈ V ↔ {𝐴} = ∅)
109biimpi 215 . . . 4 𝐴 ∈ V → {𝐴} = ∅)
1110rabeqdv 3425 . . 3 𝐴 ∈ V → {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝑥 ∈ ∅ ∣ 𝜑})
1210ifeq1d 4510 . . 3 𝐴 ∈ V → if(𝜓, {𝐴}, ∅) = if(𝜓, ∅, ∅))
138, 11, 123eqtr4a 2803 . 2 𝐴 ∈ V → {𝑥 ∈ {𝐴} ∣ 𝜑} = if(𝜓, {𝐴}, ∅))
145, 13pm2.61i 182 1 {𝑥 ∈ {𝐴} ∣ 𝜑} = if(𝜓, {𝐴}, ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205   = wceq 1542  wcel 2107  {crab 3410  Vcvv 3448  [wsbc 3744  c0 4287  ifcif 4491  {csn 4591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-rab 3411  df-v 3450  df-sbc 3745  df-dif 3918  df-un 3920  df-nul 4288  df-if 4492  df-sn 4592
This theorem is referenced by:  suppsnop  8114  m1detdiag  21962  left1s  27246  right1s  27247  1loopgrvd2  28493  1hevtxdg1  28496  1egrvtxdg1  28499
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