Theorem rabsnif 4690

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

Step	Hyp	Ref	Expression
1		rabsnifsb 4689	. . 3 ⊢ {𝑥 ∈ {𝐴} ∣ 𝜑} = if([𝐴 / 𝑥]𝜑, {𝐴}, ∅)
2		rabsnif.f	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
3	2	sbcieg 3784	. . . 4 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))
4	3	ifbid 4512	. . 3 ⊢ (𝐴 ∈ V → if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = if(𝜓, {𝐴}, ∅))
5	1, 4	eqtrid 2810	. 2 ⊢ (𝐴 ∈ V → {𝑥 ∈ {𝐴} ∣ 𝜑} = if(𝜓, {𝐴}, ∅))
6		rab0 4343	. . . 4 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = ∅
7		ifid 4529	. . . 4 ⊢ if(𝜓, ∅, ∅) = ∅
8	6, 7	eqtr4i 2789	. . 3 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = if(𝜓, ∅, ∅)
9		snprc 4684	. . . . 5 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
10	9	biimpi 219	. . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
11	10	rabeqdv 3431	. . 3 ⊢ (¬ 𝐴 ∈ V → {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝑥 ∈ ∅ ∣ 𝜑})
12	10	ifeq1d 4508	. . 3 ⊢ (¬ 𝐴 ∈ V → if(𝜓, {𝐴}, ∅) = if(𝜓, ∅, ∅))
13	8, 11, 12	3eqtr4a 2824	. 2 ⊢ (¬ 𝐴 ∈ V → {𝑥 ∈ {𝐴} ∣ 𝜑} = if(𝜓, {𝐴}, ∅))
14	5, 13	pm2.61i 184	1 ⊢ {𝑥 ∈ {𝐴} ∣ 𝜑} = if(𝜓, {𝐴}, ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2143  {crab 3416  Vcvv 3455  [wsbc 3745  ∅c0 4287  ifcif 4488  {csn 4590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-dif 3909  df-un 3911  df-nul 4288  df-if 4489  df-sn 4591

This theorem is referenced by:  suppsnop  8175  m1detdiag  22735  left1s  28069  right1s  28070  1loopgrvd2  29834  1hevtxdg1  29837  1egrvtxdg1  29840