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Theorem rabsnifsb 4688
Description: A restricted class abstraction restricted to a singleton is either the empty set or the singleton itself. (Contributed by AV, 21-Jul-2019.)
Assertion
Ref Expression
rabsnifsb {𝑥 ∈ {𝐴} ∣ 𝜑} = if([𝐴 / 𝑥]𝜑, {𝐴}, ∅)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rabsnifsb
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 elsni 4608 . . . . . . . 8 (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
2 sbceq1a 3755 . . . . . . . . 9 (𝑥 = 𝐴 → (𝜑[𝐴 / 𝑥]𝜑))
32biimpd 228 . . . . . . . 8 (𝑥 = 𝐴 → (𝜑[𝐴 / 𝑥]𝜑))
41, 3syl 17 . . . . . . 7 (𝑥 ∈ {𝐴} → (𝜑[𝐴 / 𝑥]𝜑))
54imdistani 570 . . . . . 6 ((𝑥 ∈ {𝐴} ∧ 𝜑) → (𝑥 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑))
65orcd 872 . . . . 5 ((𝑥 ∈ {𝐴} ∧ 𝜑) → ((𝑥 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) ∨ (𝑥 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑)))
72biimprd 248 . . . . . . . 8 (𝑥 = 𝐴 → ([𝐴 / 𝑥]𝜑𝜑))
81, 7syl 17 . . . . . . 7 (𝑥 ∈ {𝐴} → ([𝐴 / 𝑥]𝜑𝜑))
98imdistani 570 . . . . . 6 ((𝑥 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) → (𝑥 ∈ {𝐴} ∧ 𝜑))
10 noel 4295 . . . . . . . 8 ¬ 𝑥 ∈ ∅
1110pm2.21i 119 . . . . . . 7 (𝑥 ∈ ∅ → (𝑥 ∈ {𝐴} ∧ 𝜑))
1211adantr 482 . . . . . 6 ((𝑥 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑) → (𝑥 ∈ {𝐴} ∧ 𝜑))
139, 12jaoi 856 . . . . 5 (((𝑥 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) ∨ (𝑥 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑)) → (𝑥 ∈ {𝐴} ∧ 𝜑))
146, 13impbii 208 . . . 4 ((𝑥 ∈ {𝐴} ∧ 𝜑) ↔ ((𝑥 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) ∨ (𝑥 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑)))
1514abbii 2807 . . 3 {𝑥 ∣ (𝑥 ∈ {𝐴} ∧ 𝜑)} = {𝑥 ∣ ((𝑥 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) ∨ (𝑥 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑))}
16 nfv 1918 . . . 4 𝑦((𝑥 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) ∨ (𝑥 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑))
17 nfv 1918 . . . . . 6 𝑥 𝑦 ∈ {𝐴}
18 nfsbc1v 3764 . . . . . 6 𝑥[𝐴 / 𝑥]𝜑
1917, 18nfan 1903 . . . . 5 𝑥(𝑦 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑)
20 nfv 1918 . . . . . 6 𝑥 𝑦 ∈ ∅
2118nfn 1861 . . . . . 6 𝑥 ¬ [𝐴 / 𝑥]𝜑
2220, 21nfan 1903 . . . . 5 𝑥(𝑦 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑)
2319, 22nfor 1908 . . . 4 𝑥((𝑦 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) ∨ (𝑦 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑))
24 eleq1w 2821 . . . . . 6 (𝑥 = 𝑦 → (𝑥 ∈ {𝐴} ↔ 𝑦 ∈ {𝐴}))
2524anbi1d 631 . . . . 5 (𝑥 = 𝑦 → ((𝑥 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) ↔ (𝑦 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑)))
26 eleq1w 2821 . . . . . 6 (𝑥 = 𝑦 → (𝑥 ∈ ∅ ↔ 𝑦 ∈ ∅))
2726anbi1d 631 . . . . 5 (𝑥 = 𝑦 → ((𝑥 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑) ↔ (𝑦 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑)))
2825, 27orbi12d 918 . . . 4 (𝑥 = 𝑦 → (((𝑥 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) ∨ (𝑥 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑)) ↔ ((𝑦 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) ∨ (𝑦 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑))))
2916, 23, 28cbvabw 2811 . . 3 {𝑥 ∣ ((𝑥 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) ∨ (𝑥 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑))} = {𝑦 ∣ ((𝑦 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) ∨ (𝑦 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑))}
3015, 29eqtri 2765 . 2 {𝑥 ∣ (𝑥 ∈ {𝐴} ∧ 𝜑)} = {𝑦 ∣ ((𝑦 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) ∨ (𝑦 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑))}
31 df-rab 3411 . 2 {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ {𝐴} ∧ 𝜑)}
32 df-if 4492 . 2 if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = {𝑦 ∣ ((𝑦 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) ∨ (𝑦 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑))}
3330, 31, 323eqtr4i 2775 1 {𝑥 ∈ {𝐴} ∣ 𝜑} = if([𝐴 / 𝑥]𝜑, {𝐴}, ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397  wo 846   = wceq 1542  wcel 2107  {cab 2714  {crab 3410  [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-sbc 3745  df-dif 3918  df-nul 4288  df-if 4492  df-sn 4592
This theorem is referenced by:  rabsnif  4689  rabrsn  4690
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