Theorem rabsnt 4735

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 4664	. . 3 ⊢ 𝐵 ∈ {𝐵}
3		id 22	. . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = {𝐵} → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝐵})
4	2, 3	eleqtrrid 2839	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = {𝐵} → 𝐵 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑})
5		rabsnt.2	. . . 4 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))
6	5	elrab 3683	. . 3 ⊢ (𝐵 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ (𝐵 ∈ 𝐴 ∧ 𝜓))
7	6	simprbi 496	. 2 ⊢ (𝐵 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝜓)
8	4, 7	syl 17	1 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = {𝐵} → 𝜓)

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1540   ∈ wcel 2105  {crab 3431  Vcvv 3473  {csn 4628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-sn 4629

This theorem is referenced by:  ddemeas  33547