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Theorem clelab 2159
Description: Membership of a class variable in a class abstraction. (Contributed by NM, 23-Dec-1993.)
Assertion
Ref Expression
clelab (A {xφ} ↔ x(x = A φ))
Distinct variable group:   x,A
Allowed substitution hint:   φ(x)

Proof of Theorem clelab
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 df-clab 2024 . . . 4 (y {xφ} ↔ [y / x]φ)
21anbi2i 430 . . 3 ((y = A y {xφ}) ↔ (y = A [y / x]φ))
32exbii 1493 . 2 (y(y = A y {xφ}) ↔ y(y = A [y / x]φ))
4 df-clel 2033 . 2 (A {xφ} ↔ y(y = A y {xφ}))
5 nfv 1418 . . 3 y(x = A φ)
6 nfv 1418 . . . 4 x y = A
7 nfs1v 1812 . . . 4 x[y / x]φ
86, 7nfan 1454 . . 3 x(y = A [y / x]φ)
9 eqeq1 2043 . . . 4 (x = y → (x = Ay = A))
10 sbequ12 1651 . . . 4 (x = y → (φ ↔ [y / x]φ))
119, 10anbi12d 442 . . 3 (x = y → ((x = A φ) ↔ (y = A [y / x]φ)))
125, 8, 11cbvex 1636 . 2 (x(x = A φ) ↔ y(y = A [y / x]φ))
133, 4, 123bitr4i 201 1 (A {xφ} ↔ x(x = A φ))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   = wceq 1242  wex 1378   wcel 1390  [wsb 1642  {cab 2023
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-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033
This theorem is referenced by:  elrabi  2689
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