Theorem rabsnt 3798

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 3761	. . 3 ⊢ B ∈ {B}
3		id 19	. . 3 ⊢ ({x ∈ A ∣ φ} = {B} → {x ∈ A ∣ φ} = {B})
4	2, 3	syl5eleqr 2440	. 2 ⊢ ({x ∈ A ∣ φ} = {B} → B ∈ {x ∈ A ∣ φ})
5		rabsnt.2	. . . 4 ⊢ (x = B → (φ ↔ ψ))
6	5	elrab 2995	. . 3 ⊢ (B ∈ {x ∈ A ∣ φ} ↔ (B ∈ A ∧ ψ))
7	6	simprbi 450	. 2 ⊢ (B ∈ {x ∈ A ∣ φ} → ψ)
8	4, 7	syl 15	1 ⊢ ({x ∈ A ∣ φ} = {B} → ψ)

Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   ∈ wcel 1710  {crab 2619  Vcvv 2860  {csn 3738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rab 2624  df-v 2862  df-sn 3742

This theorem is referenced by: (None)