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Theorem elabgf 2983
 Description: Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. This version has bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.) (Revised by Mario Carneiro, 12-Oct-2016.)
Hypotheses
Ref Expression
elabgf.1 xA
elabgf.2 xψ
elabgf.3 (x = A → (φψ))
Assertion
Ref Expression
elabgf (A B → (A {x φ} ↔ ψ))

Proof of Theorem elabgf
StepHypRef Expression
1 elabgf.1 . 2 xA
2 nfab1 2491 . . . 4 x{x φ}
31, 2nfel 2497 . . 3 x A {x φ}
4 elabgf.2 . . 3 xψ
53, 4nfbi 1834 . 2 x(A {x φ} ↔ ψ)
6 eleq1 2413 . . 3 (x = A → (x {x φ} ↔ A {x φ}))
7 elabgf.3 . . 3 (x = A → (φψ))
86, 7bibi12d 312 . 2 (x = A → ((x {x φ} ↔ φ) ↔ (A {x φ} ↔ ψ)))
9 abid 2341 . 2 (x {x φ} ↔ φ)
101, 5, 8, 9vtoclgf 2913 1 (A B → (A {x φ} ↔ ψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  {cab 2339  Ⅎwnfc 2476 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:  elabf  2984  elabg  2986  elab3gf  2990  elrabf  2993
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