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Theorem elab3gf 2990
 Description: Membership in a class abstraction, with a weaker antecedent than elabgf 2983. (Contributed by NM, 6-Sep-2011.)
Hypotheses
Ref Expression
elab3gf.1 xA
elab3gf.2 xψ
elab3gf.3 (x = A → (φψ))
Assertion
Ref Expression
elab3gf ((ψA B) → (A {x φ} ↔ ψ))

Proof of Theorem elab3gf
StepHypRef Expression
1 elab3gf.1 . . . . 5 xA
2 elab3gf.2 . . . . 5 xψ
3 elab3gf.3 . . . . 5 (x = A → (φψ))
41, 2, 3elabgf 2983 . . . 4 (A {x φ} → (A {x φ} ↔ ψ))
54ibi 232 . . 3 (A {x φ} → ψ)
6 pm2.21 100 . . 3 ψ → (ψA {x φ}))
75, 6impbid2 195 . 2 ψ → (A {x φ} ↔ ψ))
81, 2, 3elabgf 2983 . 2 (A B → (A {x φ} ↔ ψ))
97, 8ja 153 1 ((ψA B) → (A {x φ} ↔ ψ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  {cab 2339  Ⅎwnfc 2476 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:  elab3g  2991
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