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Theorem elrab3 2676
Description: Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 5-Oct-2006.)
Hypothesis
Ref Expression
elrab.1 (x = A → (φψ))
Assertion
Ref Expression
elrab3 (A B → (A {x Bφ} ↔ ψ))
Distinct variable groups:   ψ,x   x,A   x,B
Allowed substitution hint:   φ(x)

Proof of Theorem elrab3
StepHypRef Expression
1 elrab.1 . . 3 (x = A → (φψ))
21elrab 2675 . 2 (A {x Bφ} ↔ (A B ψ))
32baib 818 1 (A B → (A {x Bφ} ↔ ψ))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1228   wcel 1374  {crab 2288
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
This theorem is referenced by:  unimax  3588  ordtriexmidlem2  4193  ordtriexmid  4194  ordtri2orexmid  4195  onsucelsucexmid  4199  ordpwsucexmid  4230  acexmidlema  5427  acexmidlemb  5428  genpelvl  6366  genpelvu  6367
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