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Theorem elcnv2 4429
Description: Membership in a converse. Equation 5 of [Suppes] p. 62. (Contributed by NM, 11-Aug-2004.)
Assertion
Ref Expression
elcnv2 (A 𝑅xy(A = ⟨x, yy, x 𝑅))
Distinct variable groups:   x,y,A   x,𝑅,y

Proof of Theorem elcnv2
StepHypRef Expression
1 elcnv 4428 . 2 (A 𝑅xy(A = ⟨x, y y𝑅x))
2 df-br 3729 . . . 4 (y𝑅x ↔ ⟨y, x 𝑅)
32anbi2i 430 . . 3 ((A = ⟨x, y y𝑅x) ↔ (A = ⟨x, yy, x 𝑅))
432exbii 1471 . 2 (xy(A = ⟨x, y y𝑅x) ↔ xy(A = ⟨x, yy, x 𝑅))
51, 4bitri 173 1 (A 𝑅xy(A = ⟨x, yy, x 𝑅))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   = wceq 1224  wex 1355   wcel 1367  cop 3343   class class class wbr 3728  ccnv 4260
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 614  ax-5 1310  ax-7 1311  ax-gen 1312  ax-ie1 1356  ax-ie2 1357  ax-8 1369  ax-10 1370  ax-11 1371  ax-i12 1372  ax-bnd 1373  ax-4 1374  ax-14 1379  ax-17 1393  ax-i9 1397  ax-ial 1401  ax-i5r 1402  ax-ext 1996  ax-sep 3839  ax-pow 3891  ax-pr 3908
This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1227  df-nf 1324  df-sb 1620  df-clab 2001  df-cleq 2007  df-clel 2010  df-nfc 2141  df-v 2529  df-un 2891  df-in 2893  df-ss 2900  df-pw 3326  df-sn 3346  df-pr 3347  df-op 3349  df-br 3729  df-opab 3783  df-cnv 4269
This theorem is referenced by:  cnvuni  4437
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