Theorem rabsnif 4728

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 4727	. . 3 ⊢ {𝑥 ∈ {𝐴} ∣ 𝜑} = if([𝐴 / 𝑥]𝜑, {𝐴}, ∅)
2		rabsnif.f	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
3	2	sbcieg 3818	. . . 4 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))
4	3	ifbid 4552	. . 3 ⊢ (𝐴 ∈ V → if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = if(𝜓, {𝐴}, ∅))
5	1, 4	eqtrid 2785	. 2 ⊢ (𝐴 ∈ V → {𝑥 ∈ {𝐴} ∣ 𝜑} = if(𝜓, {𝐴}, ∅))
6		rab0 4383	. . . 4 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = ∅
7		ifid 4569	. . . 4 ⊢ if(𝜓, ∅, ∅) = ∅
8	6, 7	eqtr4i 2764	. . 3 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = if(𝜓, ∅, ∅)
9		snprc 4722	. . . . 5 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
10	9	biimpi 215	. . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
11	10	rabeqdv 3448	. . 3 ⊢ (¬ 𝐴 ∈ V → {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝑥 ∈ ∅ ∣ 𝜑})
12	10	ifeq1d 4548	. . 3 ⊢ (¬ 𝐴 ∈ V → if(𝜓, {𝐴}, ∅) = if(𝜓, ∅, ∅))
13	8, 11, 12	3eqtr4a 2799	. 2 ⊢ (¬ 𝐴 ∈ V → {𝑥 ∈ {𝐴} ∣ 𝜑} = if(𝜓, {𝐴}, ∅))
14	5, 13	pm2.61i 182	1 ⊢ {𝑥 ∈ {𝐴} ∣ 𝜑} = if(𝜓, {𝐴}, ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   = wceq 1542   ∈ wcel 2107  {crab 3433  Vcvv 3475  [wsbc 3778  ∅c0 4323  ifcif 4529  {csn 4629

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 2704

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 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-nul 4324  df-if 4530  df-sn 4630

This theorem is referenced by:  suppsnop  8163  m1detdiag  22099  left1s  27389  right1s  27390  1loopgrvd2  28760  1hevtxdg1  28763  1egrvtxdg1  28766