Theorem rabsnif 3774

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 → (𝜑 ↔ 𝜓))

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

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elrabi 2979	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∈ {𝐴} ∣ 𝜑} → 𝑦 ∈ {𝐴})
2		elsni 3723	. . . . . 6 ⊢ (𝑦 ∈ {𝐴} → 𝑦 = 𝐴)
3	1, 2	syl 14	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∈ {𝐴} ∣ 𝜑} → 𝑦 = 𝐴)
4	3	19.8ad 1644	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∈ {𝐴} ∣ 𝜑} → ∃𝑦 𝑦 = 𝐴)
5		isset 2828	. . . 4 ⊢ (𝐴 ∈ V ↔ ∃𝑦 𝑦 = 𝐴)
6	4, 5	sylibr 134	. . 3 ⊢ (𝑦 ∈ {𝑥 ∈ {𝐴} ∣ 𝜑} → 𝐴 ∈ V)
7		noel 3525	. . . . . . . . 9 ⊢ ¬ 𝑦 ∈ ∅
8	7	intnan 941	. . . . . . . 8 ⊢ ¬ (¬ 𝜓 ∧ 𝑦 ∈ ∅)
9	8	a1i 9	. . . . . . 7 ⊢ (𝑦 ∈ if(𝜓, {𝐴}, ∅) → ¬ (¬ 𝜓 ∧ 𝑦 ∈ ∅))
10		elif 3649	. . . . . . . 8 ⊢ (𝑦 ∈ if(𝜓, {𝐴}, ∅) ↔ ((𝜓 ∧ 𝑦 ∈ {𝐴}) ∨ (¬ 𝜓 ∧ 𝑦 ∈ ∅)))
11	10	biimpi 120	. . . . . . 7 ⊢ (𝑦 ∈ if(𝜓, {𝐴}, ∅) → ((𝜓 ∧ 𝑦 ∈ {𝐴}) ∨ (¬ 𝜓 ∧ 𝑦 ∈ ∅)))
12	9, 11	ecased 1390	. . . . . 6 ⊢ (𝑦 ∈ if(𝜓, {𝐴}, ∅) → (𝜓 ∧ 𝑦 ∈ {𝐴}))
13	12, 2	simpl2im 390	. . . . 5 ⊢ (𝑦 ∈ if(𝜓, {𝐴}, ∅) → 𝑦 = 𝐴)
14	13	19.8ad 1644	. . . 4 ⊢ (𝑦 ∈ if(𝜓, {𝐴}, ∅) → ∃𝑦 𝑦 = 𝐴)
15	14, 5	sylibr 134	. . 3 ⊢ (𝑦 ∈ if(𝜓, {𝐴}, ∅) → 𝐴 ∈ V)
16		rabsnifsb 3773	. . . . 5 ⊢ {𝑥 ∈ {𝐴} ∣ 𝜑} = if([𝐴 / 𝑥]𝜑, {𝐴}, ∅)
17		rabsnif.f	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
18	17	sbcieg 3084	. . . . . 6 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))
19	18	ifbid 3659	. . . . 5 ⊢ (𝐴 ∈ V → if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = if(𝜓, {𝐴}, ∅))
20	16, 19	eqtrid 2283	. . . 4 ⊢ (𝐴 ∈ V → {𝑥 ∈ {𝐴} ∣ 𝜑} = if(𝜓, {𝐴}, ∅))
21	20	eleq2d 2308	. . 3 ⊢ (𝐴 ∈ V → (𝑦 ∈ {𝑥 ∈ {𝐴} ∣ 𝜑} ↔ 𝑦 ∈ if(𝜓, {𝐴}, ∅)))
22	6, 15, 21	pm5.21nii 716	. 2 ⊢ (𝑦 ∈ {𝑥 ∈ {𝐴} ∣ 𝜑} ↔ 𝑦 ∈ if(𝜓, {𝐴}, ∅))
23	22	eqriv 2235	1 ⊢ {𝑥 ∈ {𝐴} ∣ 𝜑} = if(𝜓, {𝐴}, ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 720   = wceq 1402  ∃wex 1545   ∈ wcel 2209  {crab 2532  Vcvv 2821  [wsbc 3051  ∅c0 3520  ifcif 3635  {csn 3705

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-nul 3521  df-if 3636  df-sn 3711

This theorem is referenced by:  suppsnopdc  6480  1loopgrvd2fi  16460  1hevtxdg1en  16463