Theorem rabsnif 4681

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 4680	. . 3 ⊢ {𝑥 ∈ {𝐴} ∣ 𝜑} = if([𝐴 / 𝑥]𝜑, {𝐴}, ∅)
2		rabsnif.f	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
3	2	sbcieg 3783	. . . 4 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))
4	3	ifbid 4503	. . 3 ⊢ (𝐴 ∈ V → if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = if(𝜓, {𝐴}, ∅))
5	1, 4	eqtrid 2808	. 2 ⊢ (𝐴 ∈ V → {𝑥 ∈ {𝐴} ∣ 𝜑} = if(𝜓, {𝐴}, ∅))
6		rab0 4338	. . . 4 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = ∅
7		ifid 4520	. . . 4 ⊢ if(𝜓, ∅, ∅) = ∅
8	6, 7	eqtr4i 2787	. . 3 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = if(𝜓, ∅, ∅)
9		snprc 4675	. . . . 5 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
10	9	biimpi 218	. . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
11	10	rabeqdv 3428	. . 3 ⊢ (¬ 𝐴 ∈ V → {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝑥 ∈ ∅ ∣ 𝜑})
12	10	ifeq1d 4499	. . 3 ⊢ (¬ 𝐴 ∈ V → if(𝜓, {𝐴}, ∅) = if(𝜓, ∅, ∅))
13	8, 11, 12	3eqtr4a 2822	. 2 ⊢ (¬ 𝐴 ∈ V → {𝑥 ∈ {𝐴} ∣ 𝜑} = if(𝜓, {𝐴}, ∅))
14	5, 13	pm2.61i 183	1 ⊢ {𝑥 ∈ {𝐴} ∣ 𝜑} = if(𝜓, {𝐴}, ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   = wceq 1559   ∈ wcel 2141  {crab 3413  Vcvv 3453  [wsbc 3744  ∅c0 4285  ifcif 4479  {csn 4581

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3745  df-dif 3907  df-un 3909  df-nul 4286  df-if 4480  df-sn 4582

This theorem is referenced by:  suppsnop  8153  m1detdiag  22637  left1s  27965  right1s  27966  1loopgrvd2  29650  1hevtxdg1  29653  1egrvtxdg1  29656