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Theorem rabsnif 4403
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 4402 . . 3 {𝑥 ∈ {𝐴} ∣ 𝜑} = if([𝐴 / 𝑥]𝜑, {𝐴}, ∅)
2 rabsnif.f . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
32sbcieg 3610 . . . 4 (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑𝜓))
43ifbid 4253 . . 3 (𝐴 ∈ V → if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = if(𝜓, {𝐴}, ∅))
51, 4syl5eq 2807 . 2 (𝐴 ∈ V → {𝑥 ∈ {𝐴} ∣ 𝜑} = if(𝜓, {𝐴}, ∅))
6 rab0 4099 . . . 4 {𝑥 ∈ ∅ ∣ 𝜑} = ∅
7 ifid 4270 . . . 4 if(𝜓, ∅, ∅) = ∅
86, 7eqtr4i 2786 . . 3 {𝑥 ∈ ∅ ∣ 𝜑} = if(𝜓, ∅, ∅)
9 snprc 4398 . . . . 5 𝐴 ∈ V ↔ {𝐴} = ∅)
109biimpi 206 . . . 4 𝐴 ∈ V → {𝐴} = ∅)
1110rabeqdv 3335 . . 3 𝐴 ∈ V → {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝑥 ∈ ∅ ∣ 𝜑})
1210ifeq1d 4249 . . 3 𝐴 ∈ V → if(𝜓, {𝐴}, ∅) = if(𝜓, ∅, ∅))
138, 11, 123eqtr4a 2821 . 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 1632  wcel 2140  {crab 3055  Vcvv 3341  [wsbc 3577  c0 4059  ifcif 4231  {csn 4322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-rab 3060  df-v 3343  df-sbc 3578  df-dif 3719  df-un 3721  df-nul 4060  df-if 4232  df-sn 4323
This theorem is referenced by:  suppsnop  7479  m1detdiag  20626  1loopgrvd2  26631  1hevtxdg1  26634  1egrvtxdg1  26637
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