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Theorem elab3g 2991
 Description: Membership in a class abstraction, with a weaker antecedent than elabg 2986. (Contributed by NM, 29-Aug-2006.)
Hypothesis
Ref Expression
elab3g.1 (x = A → (φψ))
Assertion
Ref Expression
elab3g ((ψA B) → (A {x φ} ↔ ψ))
Distinct variable groups:   ψ,x   x,A
Allowed substitution hints:   φ(x)   B(x)

Proof of Theorem elab3g
StepHypRef Expression
1 nfcv 2489 . 2 xA
2 nfv 1619 . 2 xψ
3 elab3g.1 . 2 (x = A → (φψ))
41, 2, 3elab3gf 2990 1 ((ψA B) → (A {x φ} ↔ ψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   ∈ wcel 1710  {cab 2339 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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:  elab3  2992
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