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Theorem elab3g 3650
 Description: Membership in a class abstraction, with a weaker antecedent than elabg 3643. (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 2974 . 2 𝑥𝐴
2 nfv 1916 . 2 𝑥𝜓
3 elab3g.1 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
41, 2, 3elab3gf 3649 1 ((𝜓𝐴𝐵) → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2115  {cab 2799 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-v 3473 This theorem is referenced by:  elab3  3651  elssabg  5212  elrnmptg  5804  elrelimasn  5926  elmapg  8394  isust  22787  ellimc  24454  isismty  35119  clublem  40107
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