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Theorem elab4g 2989
 Description: Membership in a class abstraction, using implicit substitution. (Contributed by NM, 17-Oct-2012.)
Hypotheses
Ref Expression
elab4g.1 (x = A → (φψ))
elab4g.2 B = {x φ}
Assertion
Ref Expression
elab4g (A B ↔ (A V ψ))
Distinct variable groups:   ψ,x   x,A
Allowed substitution hints:   φ(x)   B(x)

Proof of Theorem elab4g
StepHypRef Expression
1 elex 2867 . 2 (A BA V)
2 elab4g.1 . . 3 (x = A → (φψ))
3 elab4g.2 . . 3 B = {x φ}
42, 3elab2g 2987 . 2 (A V → (A Bψ))
51, 4biadan2 623 1 (A B ↔ (A V ψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339  Vcvv 2859 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861 This theorem is referenced by: (None)
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