Theorem rabsnt 3746

Description: Truth implied by equality of a restricted class abstraction and a singleton. (Contributed by NM, 29-May-2006.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)

Hypotheses

Ref	Expression
rabsnt.1	⊢ 𝐵 ∈ V
rabsnt.2	⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
rabsnt	⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = {𝐵} → 𝜓)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜓,𝑥

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rabsnt

Step	Hyp	Ref	Expression
1		rabsnt.1	. . . 4 ⊢ 𝐵 ∈ V
2	1	snid 3700	. . 3 ⊢ 𝐵 ∈ {𝐵}
3		id 19	. . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = {𝐵} → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝐵})
4	2, 3	eleqtrrid 2321	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = {𝐵} → 𝐵 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑})
5		rabsnt.2	. . . 4 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))
6	5	elrab 2962	. . 3 ⊢ (𝐵 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ (𝐵 ∈ 𝐴 ∧ 𝜓))
7	6	simprbi 275	. 2 ⊢ (𝐵 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝜓)
8	4, 7	syl 14	1 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = {𝐵} → 𝜓)

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1397   ∈ wcel 2202  {crab 2514  Vcvv 2802  {csn 3669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rab 2519  df-v 2804  df-sn 3675

This theorem is referenced by:  ontr2exmid  4623  onsucsssucexmid  4625  ordsoexmid  4660  unfiexmid  7109