Theorem rabsnifsb 4678

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.

Step	Hyp	Ref	Expression
1		elsni 4596	. . . . . . . 8 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
2		sbceq1a 3750	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑))
3	2	biimpd 229	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝜑 → [𝐴 / 𝑥]𝜑))
4	1, 3	syl 17	. . . . . . 7 ⊢ (𝑥 ∈ {𝐴} → (𝜑 → [𝐴 / 𝑥]𝜑))
5	4	imdistani 568	. . . . . 6 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝜑) → (𝑥 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑))
6	5	orcd 874	. . . . 5 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝜑) → ((𝑥 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) ∨ (𝑥 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑)))
7	2	biimprd 248	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → ([𝐴 / 𝑥]𝜑 → 𝜑))
8	1, 7	syl 17	. . . . . . 7 ⊢ (𝑥 ∈ {𝐴} → ([𝐴 / 𝑥]𝜑 → 𝜑))
9	8	imdistani 568	. . . . . 6 ⊢ ((𝑥 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) → (𝑥 ∈ {𝐴} ∧ 𝜑))
10		noel 4289	. . . . . . . 8 ⊢ ¬ 𝑥 ∈ ∅
11	10	pm2.21i 119	. . . . . . 7 ⊢ (𝑥 ∈ ∅ → (𝑥 ∈ {𝐴} ∧ 𝜑))
12	11	adantr 480	. . . . . 6 ⊢ ((𝑥 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑) → (𝑥 ∈ {𝐴} ∧ 𝜑))
13	9, 12	jaoi 858	. . . . 5 ⊢ (((𝑥 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) ∨ (𝑥 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑)) → (𝑥 ∈ {𝐴} ∧ 𝜑))
14	6, 13	impbii 209	. . . 4 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝜑) ↔ ((𝑥 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) ∨ (𝑥 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑)))
15	14	abbii 2802	. . 3 ⊢ {𝑥 ∣ (𝑥 ∈ {𝐴} ∧ 𝜑)} = {𝑥 ∣ ((𝑥 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) ∨ (𝑥 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑))}
16		nfv 1916	. . . 4 ⊢ Ⅎ𝑦((𝑥 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) ∨ (𝑥 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑))
17		nfv 1916	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 ∈ {𝐴}
18		nfsbc1v 3759	. . . . . 6 ⊢ Ⅎ𝑥[𝐴 / 𝑥]𝜑
19	17, 18	nfan 1901	. . . . 5 ⊢ Ⅎ𝑥(𝑦 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑)
20		nfv 1916	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 ∈ ∅
21	18	nfn 1859	. . . . . 6 ⊢ Ⅎ𝑥 ¬ [𝐴 / 𝑥]𝜑
22	20, 21	nfan 1901	. . . . 5 ⊢ Ⅎ𝑥(𝑦 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑)
23	19, 22	nfor 1906	. . . 4 ⊢ Ⅎ𝑥((𝑦 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) ∨ (𝑦 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑))
24		eleq1w 2818	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝐴} ↔ 𝑦 ∈ {𝐴}))
25	24	anbi1d 632	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) ↔ (𝑦 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑)))
26		eleq1w 2818	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ ∅ ↔ 𝑦 ∈ ∅))
27	26	anbi1d 632	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑) ↔ (𝑦 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑)))
28	25, 27	orbi12d 919	. . . 4 ⊢ (𝑥 = 𝑦 → (((𝑥 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) ∨ (𝑥 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑)) ↔ ((𝑦 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) ∨ (𝑦 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑))))
29	16, 23, 28	cbvabw 2806	. . 3 ⊢ {𝑥 ∣ ((𝑥 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) ∨ (𝑥 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑))} = {𝑦 ∣ ((𝑦 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) ∨ (𝑦 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑))}
30	15, 29	eqtri 2758	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ {𝐴} ∧ 𝜑)} = {𝑦 ∣ ((𝑦 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) ∨ (𝑦 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑))}
31		df-rab 3399	. 2 ⊢ {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ {𝐴} ∧ 𝜑)}
32		df-if 4479	. 2 ⊢ if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = {𝑦 ∣ ((𝑦 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) ∨ (𝑦 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑))}
33	30, 31, 32	3eqtr4i 2768	1 ⊢ {𝑥 ∈ {𝐴} ∣ 𝜑} = if([𝐴 / 𝑥]𝜑, {𝐴}, ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114  {cab 2713  {crab 3398  [wsbc 3739  ∅c0 4284  ifcif 4478  {csn 4579

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-rab 3399  df-sbc 3740  df-dif 3903  df-nul 4285  df-if 4479  df-sn 4580

This theorem is referenced by:  rabsnif  4679  rabrsn  4680