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Theorem elabgf 2679
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 {xφ} ↔ ψ))

Proof of Theorem elabgf
StepHypRef Expression
1 elabgf.1 . 2 xA
2 nfab1 2177 . . . 4 x{xφ}
31, 2nfel 2183 . . 3 x A {xφ}
4 elabgf.2 . . 3 xψ
53, 4nfbi 1478 . 2 x(A {xφ} ↔ ψ)
6 eleq1 2097 . . 3 (x = A → (x {xφ} ↔ A {xφ}))
7 elabgf.3 . . 3 (x = A → (φψ))
86, 7bibi12d 224 . 2 (x = A → ((x {xφ} ↔ φ) ↔ (A {xφ} ↔ ψ)))
9 abid 2025 . 2 (x {xφ} ↔ φ)
101, 5, 8, 9vtoclgf 2606 1 (A B → (A {xφ} ↔ ψ))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1242  wnf 1346   wcel 1390  {cab 2023  wnfc 2162
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553
This theorem is referenced by:  elabf  2680  elabg  2682  elab3gf  2686  elrabf  2690  bj-intabssel  9197
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