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Theorem sbccom2f 30499
 Description: Commutative law for double class substitution, with non free variable condition. (Contributed by Giovanni Mascellani, 31-May-2019.)
Hypotheses
Ref Expression
sbccom2f.1
sbccom2f.2
Assertion
Ref Expression
sbccom2f
Distinct variable group:   ,
Allowed substitution hints:   (,)   (,)   (,)

Proof of Theorem sbccom2f
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 sbcco 3334 . . . 4
21bicomi 202 . . 3
32sbcbii 3371 . 2
4 sbccom2f.1 . . 3
54sbccom2 30498 . 2
6 vex 3096 . . . . . . 7
76sbccom2 30498 . . . . . 6
8 sbccom2f.2 . . . . . . . 8
9 eqidd 2442 . . . . . . . 8
106, 8, 9csbief 3442 . . . . . . 7
11 dfsbcq 3313 . . . . . . 7
1210, 11ax-mp 5 . . . . . 6
137, 12bitri 249 . . . . 5
1413bicomi 202 . . . 4
1514sbcbii 3371 . . 3
16 sbcco 3334 . . 3
1715, 16bitri 249 . 2
183, 5, 173bitri 271 1
 Colors of variables: wff setvar class Syntax hints:   wb 184   wceq 1381   wcel 1802  wnfc 2589  cvv 3093  wsbc 3311  csb 3417 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-v 3095  df-sbc 3312  df-csb 3418 This theorem is referenced by:  sbccom2fi  30500
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