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Theorem frege116 36646
 Description: One direction of dffrege115 36645. Proposition 116 of [Frege1879] p. 77. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
frege116.x
Assertion
Ref Expression
frege116
Distinct variable groups:   ,,   ,,
Allowed substitution hints:   (,)

Proof of Theorem frege116
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 dffrege115 36645 . 2
2 frege116.x . . . 4
32frege68c 36598 . . 3
4 sbcal 3305 . . . 4
5 sbcimg 3297 . . . . . . 7
62, 5ax-mp 5 . . . . . 6
7 sbcbr2g 4451 . . . . . . . . 9
82, 7ax-mp 5 . . . . . . . 8
9 csbvarg 3796 . . . . . . . . . 10
102, 9ax-mp 5 . . . . . . . . 9
1110breq2i 4403 . . . . . . . 8
128, 11bitri 257 . . . . . . 7
13 sbcal 3305 . . . . . . . 8
14 sbcimg 3297 . . . . . . . . . . 11
152, 14ax-mp 5 . . . . . . . . . 10
16 sbcg 3321 . . . . . . . . . . . 12
172, 16ax-mp 5 . . . . . . . . . . 11
18 sbceq2g 3783 . . . . . . . . . . . . 13
192, 18ax-mp 5 . . . . . . . . . . . 12
2010eqeq2i 2483 . . . . . . . . . . . 12
2119, 20bitri 257 . . . . . . . . . . 11
2217, 21imbi12i 333 . . . . . . . . . 10
2315, 22bitri 257 . . . . . . . . 9
2423albii 1699 . . . . . . . 8
2513, 24bitri 257 . . . . . . 7
2612, 25imbi12i 333 . . . . . 6
276, 26bitri 257 . . . . 5
2827albii 1699 . . . 4
294, 28bitri 257 . . 3
303, 29syl6ib 234 . 2
311, 30ax-mp 5 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 189  wal 1450   wceq 1452   wcel 1904  wsbc 3255  csb 3349   class class class wbr 4395  ccnv 4838   wfun 5583 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-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639  ax-frege1 36457  ax-frege2 36458  ax-frege8 36476  ax-frege52a 36524  ax-frege58b 36568 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-ifp 984  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-fun 5591 This theorem is referenced by:  frege117  36647
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