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Theorem elab3g 2687
Description: Membership in a class abstraction, with a weaker antecedent than elabg 2682. (Contributed by NM, 29-Aug-2006.)
Hypothesis
Ref Expression
elab3g.1 (x = A → (φψ))
Assertion
Ref Expression
elab3g ((ψA B) → (A {xφ} ↔ ψ))
Distinct variable groups:   ψ,x   x,A
Allowed substitution hints:   φ(x)   B(x)

Proof of Theorem elab3g
StepHypRef Expression
1 nfcv 2175 . 2 xA
2 nfv 1418 . 2 xψ
3 elab3g.1 . 2 (x = A → (φψ))
41, 2, 3elab3gf 2686 1 ((ψA B) → (A {xφ} ↔ ψ))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1242   wcel 1390  {cab 2023
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-v 2553
This theorem is referenced by:  elab3  2688  elssabg  3893  elrnmptg  4529  elreimasng  4634  fvelrnb  5164
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