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Theorem sbccom2 30121
 Description: Commutative law for double class substitution. (Contributed by Giovanni Mascellani, 31-May-2019.)
Hypothesis
Ref Expression
sbccom2.1
Assertion
Ref Expression
sbccom2
Distinct variable groups:   ,   ,
Allowed substitution hints:   (,)   ()   (,)

Proof of Theorem sbccom2
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sbcco 3347 . . . . . . . 8
21bicomi 202 . . . . . . 7
32sbcbii 3384 . . . . . 6
4 sbcco 3347 . . . . . . 7
54bicomi 202 . . . . . 6
63, 5bitri 249 . . . . 5
7 vex 3109 . . . . . . 7
87sbccom2lem 30120 . . . . . 6
98sbcbii 3384 . . . . 5
106, 9bitri 249 . . . 4
11 sbccom2.1 . . . . 5
1211sbccom2lem 30120 . . . 4
13 sbcco 3347 . . . . 5
1413sbcbii 3384 . . . 4
1510, 12, 143bitri 271 . . 3
16 csbco 3438 . . . 4
17 dfsbcq 3326 . . . 4
1816, 17ax-mp 5 . . 3
1915, 18bitri 249 . 2
20 sbccom 3404 . . 3
2120sbcbii 3384 . 2
22 sbcco 3347 . 2
2319, 21, 223bitri 271 1
 Colors of variables: wff setvar class Syntax hints:   wb 184   wceq 1374   wcel 1762  cvv 3106  wsbc 3324  csb 3428 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 1961  ax-ext 2438 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 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-v 3108  df-sbc 3325  df-csb 3429 This theorem is referenced by:  sbccom2f  30122
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