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Theorem csbcomg 3276
 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 2966 . 2
2 elex 2966 . 2
3 sbccom 3234 . . . . . 6
43a1i 11 . . . . 5
5 sbcel2g 3274 . . . . . . 7
65sbcbidv 3217 . . . . . 6
76adantl 454 . . . . 5
8 sbcel2g 3274 . . . . . . 7
98sbcbidv 3217 . . . . . 6
109adantr 453 . . . . 5
114, 7, 103bitr3d 276 . . . 4
12 sbcel2g 3274 . . . . 5
1312adantr 453 . . . 4
14 sbcel2g 3274 . . . . 5
1514adantl 454 . . . 4
1611, 13, 153bitr3d 276 . . 3
1716eqrdv 2436 . 2
181, 2, 17syl2an 465 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wa 360   wceq 1653   wcel 1726  cvv 2958  wsbc 3163  csb 3253 This theorem is referenced by:  ovmpt2s  6199 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-sbc 3164  df-csb 3254
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