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Theorem rabsnt 3797
 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 → (φψ))
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
StepHypRef Expression
1 rabsnt.1 . . . 4 B V
21snid 3760 . . 3 B {B}
3 id 19 . . 3 ({x A φ} = {B} → {x A φ} = {B})
42, 3syl5eleqr 2440 . 2 ({x A φ} = {B} → B {x A φ})
5 rabsnt.2 . . . 4 (x = B → (φψ))
65elrab 2994 . . 3 (B {x A φ} ↔ (B A ψ))
76simprbi 450 . 2 (B {x A φ} → ψ)
84, 7syl 15 1 ({x A φ} = {B} → ψ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   ∈ wcel 1710  {crab 2618  Vcvv 2859  {csn 3737 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2478  df-rab 2623  df-v 2861  df-sn 3741 This theorem is referenced by: (None)
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