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Theorem elab3g 3508
Description: Membership in a class abstraction, with a weaker antecedent than elabg 3502. (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 2913 . 2 𝑥𝐴
2 nfv 1995 . 2 𝑥𝜓
3 elab3g.1 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
41, 2, 3elab3gf 3507 1 ((𝜓𝐴𝐵) → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1631  wcel 2145  {cab 2757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353
This theorem is referenced by:  elab3  3509  elssabg  4950  elrnmptg  5513  elrelimasn  5630  elmapg  8022  isust  22227  ellimc  23857  isismty  33932  clublem  38443
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