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Theorem bj-sbeq 31571
 Description: Distribute proper substitution through an equality relation. (See sbceqg 3777). (Contributed by BJ, 6-Oct-2018.)
Assertion
Ref Expression
bj-sbeq

Proof of Theorem bj-sbeq
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 dfcleq 2465 . . . . 5
21sbbii 1812 . . . 4
3 sbsbc 3259 . . . 4
4 sbcal 3305 . . . 4
52, 3, 43bitri 279 . . 3
6 vex 3034 . . . . 5
7 sbcbig 3300 . . . . 5
86, 7ax-mp 5 . . . 4
98albii 1699 . . 3
10 sbcel2 3782 . . . . 5
11 sbcel2 3782 . . . . 5
1210, 11bibi12i 322 . . . 4
1312albii 1699 . . 3
145, 9, 133bitri 279 . 2
15 dfcleq 2465 . 2
1614, 15bitr4i 260 1
 Colors of variables: wff setvar class Syntax hints:   wb 189  wal 1450   wceq 1452  wsb 1805   wcel 1904  cvv 3031  wsbc 3255  csb 3349 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-in 3397  df-ss 3404  df-nul 3723 This theorem is referenced by: (None)
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