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Theorem csbcomgOLD 3833
 Description: Commutative law for double substitution into a class. (Contributed by NM, 14-Nov-2005.) Obsolete as of 18-Aug-2018. Use csbcom 3832 instead. (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
csbcomgOLD
Distinct variable groups:   ,   ,   ,
Allowed substitution hints:   ()   ()   (,)   (,)   (,)

Proof of Theorem csbcomgOLD
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elex 3117 . 2
2 elex 3117 . 2
3 sbccom 3406 . . . . . 6
43a1i 11 . . . . 5
5 sbcel2gOLD 3827 . . . . . . 7
65sbcbidv 3385 . . . . . 6
76adantl 466 . . . . 5
8 sbcel2gOLD 3827 . . . . . . 7
98sbcbidv 3385 . . . . . 6
109adantr 465 . . . . 5
114, 7, 103bitr3d 283 . . . 4
12 sbcel2gOLD 3827 . . . . 5
1312adantr 465 . . . 4
14 sbcel2gOLD 3827 . . . . 5
1514adantl 466 . . . 4
1611, 13, 153bitr3d 283 . . 3
1716eqrdv 2459 . 2
181, 2, 17syl2an 477 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   wceq 1374   wcel 1762  cvv 3108  wsbc 3326  csb 3430 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-v 3110  df-sbc 3327  df-csb 3431 This theorem is referenced by: (None)
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