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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  |-  X  e.  V
Assertion
Ref Expression
frege70  |-  ( R hereditary  A  ->  ( X  e.  A  ->  A. y
( X R y  ->  y  e.  A
) ) )
Distinct variable groups:    y, A    y, R    y, X
Allowed substitution hint:    V( y)

Proof of Theorem frege70
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dffrege69 36599 . 2  |-  ( A. x ( x  e.  A  ->  A. y
( x R y  ->  y  e.  A
) )  <->  R hereditary  A )
2 frege70.x . . . 4  |-  X  e.  V
32frege68c 36598 . . 3  |-  ( ( A. x ( x  e.  A  ->  A. y
( x R y  ->  y  e.  A
) )  <->  R hereditary  A )  ->  ( R hereditary  A  ->  [. X  /  x ]. ( x  e.  A  ->  A. y ( x R y  ->  y  e.  A ) ) ) )
4 sbcel1v 3314 . . . . 5  |-  ( [. X  /  x ]. x  e.  A  <->  X  e.  A
)
54biimpri 211 . . . 4  |-  ( X  e.  A  ->  [. X  /  x ]. x  e.  A )
6 sbcim1 3302 . . . 4  |-  ( [. X  /  x ]. (
x  e.  A  ->  A. y ( x R y  ->  y  e.  A ) )  -> 
( [. X  /  x ]. x  e.  A  ->  [. X  /  x ]. A. y ( x R y  ->  y  e.  A ) ) )
7 sbcal 3305 . . . . 5  |-  ( [. X  /  x ]. A. y ( x R y  ->  y  e.  A )  <->  A. y [. X  /  x ]. ( x R y  ->  y  e.  A
) )
8 sbcim1 3302 . . . . . . 7  |-  ( [. X  /  x ]. (
x R y  -> 
y  e.  A )  ->  ( [. X  /  x ]. x R y  ->  [. X  /  x ]. y  e.  A
) )
9 sbcbr1g 4450 . . . . . . . . 9  |-  ( X  e.  V  ->  ( [. X  /  x ]. x R y  <->  [_ X  /  x ]_ x R y ) )
102, 9ax-mp 5 . . . . . . . 8  |-  ( [. X  /  x ]. x R y  <->  [_ X  /  x ]_ x R y )
11 csbvarg 3796 . . . . . . . . . 10  |-  ( X  e.  V  ->  [_ X  /  x ]_ x  =  X )
122, 11ax-mp 5 . . . . . . . . 9  |-  [_ X  /  x ]_ x  =  X
1312breq1i 4402 . . . . . . . 8  |-  ( [_ X  /  x ]_ x R y  <->  X R
y )
1410, 13bitri 257 . . . . . . 7  |-  ( [. X  /  x ]. x R y  <->  X R
y )
15 sbcg 3321 . . . . . . . 8  |-  ( X  e.  V  ->  ( [. X  /  x ]. y  e.  A  <->  y  e.  A ) )
162, 15ax-mp 5 . . . . . . 7  |-  ( [. X  /  x ]. y  e.  A  <->  y  e.  A
)
178, 14, 163imtr3g 277 . . . . . 6  |-  ( [. X  /  x ]. (
x R y  -> 
y  e.  A )  ->  ( X R y  ->  y  e.  A ) )
1817alimi 1692 . . . . 5  |-  ( A. y [. X  /  x ]. ( x R y  ->  y  e.  A
)  ->  A. y
( X R y  ->  y  e.  A
) )
197, 18sylbi 200 . . . 4  |-  ( [. X  /  x ]. A. y ( x R y  ->  y  e.  A )  ->  A. y
( X R y  ->  y  e.  A
) )
205, 6, 19syl56 34 . . 3  |-  ( [. X  /  x ]. (
x  e.  A  ->  A. y ( x R y  ->  y  e.  A ) )  -> 
( X  e.  A  ->  A. y ( X R y  ->  y  e.  A ) ) )
213, 20syl6 33 . 2  |-  ( ( A. x ( x  e.  A  ->  A. y
( x R y  ->  y  e.  A
) )  <->  R hereditary  A )  ->  ( R hereditary  A  -> 
( X  e.  A  ->  A. y ( X R y  ->  y  e.  A ) ) ) )
221, 21ax-mp 5 1  |-  ( R hereditary  A  ->  ( X  e.  A  ->  A. y
( X R y  ->  y  e.  A
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189   A.wal 1450    = wceq 1452    e. 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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