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Theorem elab3g 2693
 Description: Membership in a class abstraction, with a weaker antecedent than elabg 2688. (Contributed by NM, 29-Aug-2006.)
Hypothesis
Ref Expression
elab3g.1 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
elab3g ((𝜓𝐴𝐵) → (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 2178 . 2 𝑥𝐴
2 nfv 1421 . 2 𝑥𝜓
3 elab3g.1 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
41, 2, 3elab3gf 2692 1 ((𝜓𝐴𝐵) → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1243   ∈ wcel 1393  {cab 2026 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559 This theorem is referenced by:  elab3  2694  elssabg  3902  elrnmptg  4586  elreimasng  4691  fvelrnb  5221
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