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Theorem elima3 4618
Description: Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 14-Aug-1994.)
Hypothesis
Ref Expression
elima.1 A V
Assertion
Ref Expression
elima3 (A (B𝐶) ↔ x(x 𝐶 x, A B))
Distinct variable groups:   x,A   x,B   x,𝐶

Proof of Theorem elima3
StepHypRef Expression
1 elima.1 . . 3 A V
21elima2 4617 . 2 (A (B𝐶) ↔ x(x 𝐶 xBA))
3 df-br 3756 . . . 4 (xBA ↔ ⟨x, A B)
43anbi2i 430 . . 3 ((x 𝐶 xBA) ↔ (x 𝐶 x, A B))
54exbii 1493 . 2 (x(x 𝐶 xBA) ↔ x(x 𝐶 x, A B))
62, 5bitri 173 1 (A (B𝐶) ↔ x(x 𝐶 x, A B))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98  wex 1378   wcel 1390  Vcvv 2551  cop 3370   class class class wbr 3755  cima 4291
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-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301
This theorem is referenced by:  cnvresima  4753  imaiun  5342
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