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Theorem splint 25151
 Description: Splitting an intersection. (Contributed by FL, 15-Oct-2012.)
Assertion
Ref Expression
splint
Distinct variable groups:   ,   ,
Allowed substitution hint:   ()

Proof of Theorem splint
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 vex 2804 . . . 4
2 eliin 3926 . . . 4
31, 2mp1i 11 . . 3
4 undif 3547 . . . . . 6
5 equncom 3333 . . . . . . . 8
65biimpi 186 . . . . . . 7
76eqcoms 2299 . . . . . 6
84, 7sylbi 187 . . . . 5
98raleqdv 2755 . . . 4
10 ralunb 3369 . . . 4
119, 10syl6bb 252 . . 3
12 eliin 3926 . . . . . . . 8
131, 12ax-mp 8 . . . . . . 7
1413bicomi 193 . . . . . 6
1514a1i 10 . . . . 5
16 eliin 3926 . . . . . . 7
171, 16mp1i 11 . . . . . 6
1817bicomd 192 . . . . 5
1915, 18anbi12d 691 . . . 4
20 elin 3371 . . . 4
2119, 20syl6bbr 254 . . 3
223, 11, 213bitrd 270 . 2
2322eqrdv 2294 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176   wa 358   wceq 1632   wcel 1696  wral 2556  cvv 2801   cdif 3162   cun 3163   cin 3164   wss 3165  ciin 3922 This theorem is referenced by:  splintx  25152 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-iin 3924
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