Theorem rabsnt 3741

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 3697	. . 3 ⊢ 𝐵 ∈ {𝐵}
3		id 19	. . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = {𝐵} → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝐵})
4	2, 3	eleqtrrid 2319	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = {𝐵} → 𝐵 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑})
5		rabsnt.2	. . . 4 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))
6	5	elrab 2959	. . 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 1395   ∈ wcel 2200  {crab 2512  Vcvv 2799  {csn 3666

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rab 2517  df-v 2801  df-sn 3672

This theorem is referenced by:  ontr2exmid  4616  onsucsssucexmid  4618  ordsoexmid  4653  unfiexmid  7076