MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnveqb Structured version   Visualization version   GIF version

Theorem cnveqb 5493
Description: Equality theorem for converse. (Contributed by FL, 19-Sep-2011.)
Assertion
Ref Expression
cnveqb ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵𝐴 = 𝐵))

Proof of Theorem cnveqb
StepHypRef Expression
1 cnveq 5205 . 2 (𝐴 = 𝐵𝐴 = 𝐵)
2 dfrel2 5487 . . . 4 (Rel 𝐴𝐴 = 𝐴)
3 dfrel2 5487 . . . . . . 7 (Rel 𝐵𝐵 = 𝐵)
4 cnveq 5205 . . . . . . . . 9 (𝐴 = 𝐵𝐴 = 𝐵)
5 eqeq2 2620 . . . . . . . . 9 (𝐵 = 𝐵 → (𝐴 = 𝐵𝐴 = 𝐵))
64, 5syl5ibr 234 . . . . . . . 8 (𝐵 = 𝐵 → (𝐴 = 𝐵𝐴 = 𝐵))
76eqcoms 2617 . . . . . . 7 (𝐵 = 𝐵 → (𝐴 = 𝐵𝐴 = 𝐵))
83, 7sylbi 205 . . . . . 6 (Rel 𝐵 → (𝐴 = 𝐵𝐴 = 𝐵))
9 eqeq1 2613 . . . . . . 7 (𝐴 = 𝐴 → (𝐴 = 𝐵𝐴 = 𝐵))
109imbi2d 328 . . . . . 6 (𝐴 = 𝐴 → ((𝐴 = 𝐵𝐴 = 𝐵) ↔ (𝐴 = 𝐵𝐴 = 𝐵)))
118, 10syl5ibr 234 . . . . 5 (𝐴 = 𝐴 → (Rel 𝐵 → (𝐴 = 𝐵𝐴 = 𝐵)))
1211eqcoms 2617 . . . 4 (𝐴 = 𝐴 → (Rel 𝐵 → (𝐴 = 𝐵𝐴 = 𝐵)))
132, 12sylbi 205 . . 3 (Rel 𝐴 → (Rel 𝐵 → (𝐴 = 𝐵𝐴 = 𝐵)))
1413imp 443 . 2 ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵𝐴 = 𝐵))
151, 14impbid2 214 1 ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  ccnv 5026  Rel wrel 5032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-xp 5033  df-rel 5034  df-cnv 5035
This theorem is referenced by:  cnveq0  5494  weisoeq2  6483  relexpaddg  13589  relexpaddss  36812
  Copyright terms: Public domain W3C validator