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Theorem elrab3 2693
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 2692 . 2 (A {x Bφ} ↔ (A B ψ))
32baib 827 1 (A B → (A {x Bφ} ↔ ψ))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1242   wcel 1390  {crab 2304
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-bnd 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
This theorem is referenced by:  unimax  3605  ordtriexmidlem2  4209  ordtriexmid  4210  ordtri2orexmid  4211  onsucelsucexmid  4215  ordpwsucexmid  4246  acexmidlema  5446  acexmidlemb  5447  genpelvl  6494  genpelvu  6495  ublbneg  8284  negm  8286
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