ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rabsnt Structured version   GIF version

Theorem rabsnt 3419
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 3377 . . 3 B {B}
3 id 19 . . 3 ({x Aφ} = {B} → {x Aφ} = {B})
42, 3syl5eleqr 2109 . 2 ({x Aφ} = {B} → B {x Aφ})
5 rabsnt.2 . . . 4 (x = B → (φψ))
65elrab 2675 . . 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 1228   wcel 1374  {crab 2288  Vcvv 2535  {csn 3350
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rab 2293  df-v 2537  df-sn 3356
This theorem is referenced by:  onsucsssucexmid  4196  ordsoexmid  4224
  Copyright terms: Public domain W3C validator