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

Theorem sbcfung 5616
Description: Distribute proper substitution through the function predicate. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
Assertion
Ref Expression
sbcfung

Proof of Theorem sbcfung
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sbcan 3370 . . 3
2 sbcrel 5094 . . . 4
3 sbcal 3379 . . . . 5
4 sbcex2 3381 . . . . . . 7
5 sbcal 3379 . . . . . . . . 9
6 sbcimg 3369 . . . . . . . . . . 11
7 sbcbr123 4503 . . . . . . . . . . . . 13
8 csbconstg 3447 . . . . . . . . . . . . . 14
9 csbconstg 3447 . . . . . . . . . . . . . 14
108, 9breq12d 4465 . . . . . . . . . . . . 13
117, 10syl5bb 257 . . . . . . . . . . . 12
12 sbcg 3401 . . . . . . . . . . . 12
1311, 12imbi12d 320 . . . . . . . . . . 11
146, 13bitrd 253 . . . . . . . . . 10
1514albidv 1713 . . . . . . . . 9
165, 15syl5bb 257 . . . . . . . 8
1716exbidv 1714 . . . . . . 7
184, 17syl5bb 257 . . . . . 6
1918albidv 1713 . . . . 5
203, 19syl5bb 257 . . . 4
212, 20anbi12d 710 . . 3
221, 21syl5bb 257 . 2
23 dffun3 5604 . . 3
2423sbcbii 3387 . 2
25 dffun3 5604 . 2
2622, 24, 253bitr4g 288 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  E.wex 1612  e.wcel 1818  [.wsbc 3327  [_csb 3434   class class class wbr 4452  Relwrel 5009  Funwfun 5587
This theorem is referenced by:  sbcfng  5733
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-opab 4511  df-id 4800  df-rel 5011  df-cnv 5012  df-co 5013  df-fun 5595
  Copyright terms: Public domain W3C validator