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 3832	. . . 4 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))
4	3	ifbid 4554	. . 3 ⊢ (𝐴 ∈ V → if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = if(𝜓, {𝐴}, ∅))
5	1, 4	eqtrid 2787	. 2 ⊢ (𝐴 ∈ V → {𝑥 ∈ {𝐴} ∣ 𝜑} = if(𝜓, {𝐴}, ∅))
6		rab0 4392	. . . 4 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = ∅
7		ifid 4571	. . . 4 ⊢ if(𝜓, ∅, ∅) = ∅
8	6, 7	eqtr4i 2766	. . 3 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = if(𝜓, ∅, ∅)
9		snprc 4722	. . . . 5 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
10	9	biimpi 216	. . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
11	10	rabeqdv 3449	. . 3 ⊢ (¬ 𝐴 ∈ V → {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝑥 ∈ ∅ ∣ 𝜑})
12	10	ifeq1d 4550	. . 3 ⊢ (¬ 𝐴 ∈ V → if(𝜓, {𝐴}, ∅) = if(𝜓, ∅, ∅))
13	8, 11, 12	3eqtr4a 2801	. 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 206   = wceq 1537   ∈ wcel 2106  {crab 3433  Vcvv 3478  [wsbc 3791  ∅c0 4339  ifcif 4531  {csn 4631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-nul 4340  df-if 4532  df-sn 4632

This theorem is referenced by:  suppsnop  8202  m1detdiag  22619  left1s  27948  right1s  27949  1loopgrvd2  29536  1hevtxdg1  29539  1egrvtxdg1  29542