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Theorem csbcomg 3159
 Description: Commutative law for double substitution into a class. (Contributed by NM, 14-Nov-2005.)
Assertion
Ref Expression
csbcomg
Distinct variable groups:   ,   ,   ,
Allowed substitution hints:   ()   ()   (,)   (,)   (,)

Proof of Theorem csbcomg
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elex 2867 . 2
2 elex 2867 . 2
3 sbccom 3117 . . . . . 6
43a1i 10 . . . . 5
5 sbcel2g 3157 . . . . . . 7
65sbcbidv 3100 . . . . . 6
76adantl 452 . . . . 5
8 sbcel2g 3157 . . . . . . 7
98sbcbidv 3100 . . . . . 6
109adantr 451 . . . . 5
114, 7, 103bitr3d 274 . . . 4
12 sbcel2g 3157 . . . . 5
1312adantr 451 . . . 4
14 sbcel2g 3157 . . . . 5
1514adantl 452 . . . 4
1611, 13, 153bitr3d 274 . . 3
1716eqrdv 2351 . 2
181, 2, 17syl2an 463 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 176   wa 358   wceq 1642   wcel 1710  cvv 2859  wsbc 3046  csb 3136 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-sbc 3047  df-csb 3137 This theorem is referenced by: (None)
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