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Theorem elima3 4598
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 4597 . 2 (A (B𝐶) ↔ x(x 𝐶 xBA))
3 df-br 3735 . . . 4 (xBA ↔ ⟨x, A B)
43anbi2i 433 . . 3 ((x 𝐶 xBA) ↔ (x 𝐶 x, A B))
54exbii 1474 . 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 1358   wcel 1370  Vcvv 2531  cop 3349   class class class wbr 3734  cima 4271
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-br 3735  df-opab 3789  df-xp 4274  df-cnv 4276  df-dm 4278  df-rn 4279  df-res 4280  df-ima 4281
This theorem is referenced by:  cnvresima  4733  imaiun  5320
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