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Theorem rabsnt 3436
 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 Aφ} = {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 3394 . . 3 B {B}
3 id 19 . . 3 ({x Aφ} = {B} → {x Aφ} = {B})
42, 3syl5eleqr 2124 . 2 ({x Aφ} = {B} → B {x Aφ})
5 rabsnt.2 . . . 4 (x = B → (φψ))
65elrab 2692 . . 3 (B {x Aφ} ↔ (B A ψ))
76simprbi 260 . 2 (B {x Aφ} → ψ)
84, 7syl 14 1 ({x Aφ} = {B} → ψ)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1242   ∈ wcel 1390  {crab 2304  Vcvv 2551  {csn 3367 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rab 2309  df-v 2553  df-sn 3373 This theorem is referenced by:  onsucsssucexmid  4212  ordsoexmid  4240
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