Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  riotaxfrd Structured version   Unicode version

Theorem riotaxfrd 6294
 Description: Change the variable in the expression for "the unique such that " to another variable contained in expression . Use reuhypd 4645 to eliminate the last hypothesis. (Contributed by NM, 16-Jan-2012.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
riotaxfrd.1
riotaxfrd.2
riotaxfrd.3
riotaxfrd.4
riotaxfrd.5
riotaxfrd.6
Assertion
Ref Expression
riotaxfrd
Distinct variable groups:   ,   ,   ,,   ,,   ,   ,
Allowed substitution hints:   ()   ()   ()   ()

Proof of Theorem riotaxfrd
StepHypRef Expression
1 rabid 3005 . . . 4
21baib 911 . . 3
32riotabiia 6281 . 2
4 riotaxfrd.2 . . . . . 6
5 riotaxfrd.6 . . . . . 6
6 riotaxfrd.4 . . . . . 6
74, 5, 6reuxfrd 4643 . . . . 5
8 riotacl2 6277 . . . . . . . 8
98adantl 467 . . . . . . 7
10 riotacl 6278 . . . . . . . 8
11 nfriota1 6271 . . . . . . . . 9
12 riotaxfrd.1 . . . . . . . . 9
13 riotaxfrd.5 . . . . . . . . 9
1411, 12, 4, 6, 13rabxfrd 4639 . . . . . . . 8
1510, 14sylan2 476 . . . . . . 7
169, 15mpbird 235 . . . . . 6
1716ex 435 . . . . 5
187, 17sylbid 218 . . . 4
1918imp 430 . . 3
20 riotaxfrd.3 . . . . . . . 8
2120ex 435 . . . . . . 7
2210, 21syl5 33 . . . . . 6
237, 22sylbid 218 . . . . 5
2423imp 430 . . . 4
251baibr 912 . . . . . . 7
2625reubiia 3014 . . . . . 6
2726biimpi 197 . . . . 5
2827adantl 467 . . . 4
29 nfcv 2584 . . . . 5
30 nfrab1 3009 . . . . . 6
3130nfel2 2602 . . . . 5
32 eleq1 2494 . . . . 5
3329, 31, 32riota2f 6285 . . . 4
3424, 28, 33syl2anc 665 . . 3
3519, 34mpbid 213 . 2
363, 35syl5eqr 2477 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 187   wa 370   wceq 1437   wcel 1868  wnfc 2570  wreu 2777  crab 2779  crio 6263 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-un 3441  df-in 3443  df-ss 3450  df-sn 3997  df-pr 3999  df-uni 4217  df-iota 5562  df-riota 6264 This theorem is referenced by:  riotaneg  10587  zriotaneg  11050  riotaocN  32694
 Copyright terms: Public domain W3C validator