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Theorem sbcabel 3383
 Description: Interchange class substitution and class abstraction. (Contributed by NM, 5-Nov-2005.)
Hypothesis
Ref Expression
sbcabel.1
Assertion
Ref Expression
sbcabel
Distinct variable groups:   ,   ,
Allowed substitution hints:   (,)   ()   (,)   (,)

Proof of Theorem sbcabel
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elex 3087 . 2
2 sbcex2 3348 . . . 4
3 sbcan 3337 . . . . . 6
4 sbcal 3346 . . . . . . . . 9
5 sbcbig 3341 . . . . . . . . . . 11
6 sbcg 3368 . . . . . . . . . . . 12
76bibi1d 319 . . . . . . . . . . 11
85, 7bitrd 253 . . . . . . . . . 10
98albidv 1680 . . . . . . . . 9
104, 9syl5bb 257 . . . . . . . 8
11 abeq2 2578 . . . . . . . . 9
1211sbcbii 3354 . . . . . . . 8
13 abeq2 2578 . . . . . . . 8
1410, 12, 133bitr4g 288 . . . . . . 7
15 sbcabel.1 . . . . . . . . 9
1615nfcri 2609 . . . . . . . 8
1716sbcgf 3366 . . . . . . 7
1814, 17anbi12d 710 . . . . . 6
193, 18syl5bb 257 . . . . 5
2019exbidv 1681 . . . 4
212, 20syl5bb 257 . . 3
22 df-clel 2449 . . . 4
2322sbcbii 3354 . . 3
24 df-clel 2449 . . 3
2521, 23, 243bitr4g 288 . 2
261, 25syl 16 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369  wal 1368   wceq 1370  wex 1587   wcel 1758  cab 2439  wnfc 2602  cvv 3078  wsbc 3294 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3080  df-sbc 3295 This theorem is referenced by:  csbexg  4535  csbexgOLD  4537
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