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Theorem frege70 36600
 Description: Lemma for frege72 36602. Proposition 70 of [Frege1879] p. 58. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 3-Jul-2020.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
frege70.x
Assertion
Ref Expression
frege70 hereditary
Distinct variable groups:   ,   ,   ,
Allowed substitution hint:   ()

Proof of Theorem frege70
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 dffrege69 36599 . 2 hereditary
2 frege70.x . . . 4
32frege68c 36598 . . 3 hereditary hereditary
4 sbcel1v 3314 . . . . 5
54biimpri 211 . . . 4
6 sbcim1 3302 . . . 4
7 sbcal 3305 . . . . 5
8 sbcim1 3302 . . . . . . 7
9 sbcbr1g 4450 . . . . . . . . 9
102, 9ax-mp 5 . . . . . . . 8
11 csbvarg 3796 . . . . . . . . . 10
122, 11ax-mp 5 . . . . . . . . 9
1312breq1i 4402 . . . . . . . 8
1410, 13bitri 257 . . . . . . 7
15 sbcg 3321 . . . . . . . 8
162, 15ax-mp 5 . . . . . . 7
178, 14, 163imtr3g 277 . . . . . 6
1817alimi 1692 . . . . 5
197, 18sylbi 200 . . . 4
205, 6, 19syl56 34 . . 3
213, 20syl6 33 . 2 hereditary hereditary
221, 21ax-mp 5 1 hereditary
 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   hereditary whe 36438 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-xp 4845  df-cnv 4847  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-he 36439 This theorem is referenced by:  frege71  36601
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