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Theorem frege72 36602
Description: If property  A is hereditary in the  R-sequence, if  x has property  A, and if  y is a result of an application of the procedure  R to  x, then  y has property  A. Proposition 72 of [Frege1879] p. 59. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
frege72.x  |-  X  e.  U
frege72.y  |-  Y  e.  V
Assertion
Ref Expression
frege72  |-  ( R hereditary  A  ->  ( X  e.  A  ->  ( X R Y  ->  Y  e.  A ) ) )

Proof of Theorem frege72
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 frege72.y . . . 4  |-  Y  e.  V
21frege58c 36588 . . 3  |-  ( A. z ( X R z  ->  z  e.  A )  ->  [. Y  /  z ]. ( X R z  ->  z  e.  A ) )
3 sbcim1 3302 . . . 4  |-  ( [. Y  /  z ]. ( X R z  ->  z  e.  A )  ->  ( [. Y  /  z ]. X R z  ->  [. Y  /  z ]. z  e.  A
) )
4 sbcbr2g 4451 . . . . . 6  |-  ( Y  e.  V  ->  ( [. Y  /  z ]. X R z  <->  X R [_ Y  /  z ]_ z ) )
5 csbvarg 3796 . . . . . . 7  |-  ( Y  e.  V  ->  [_ Y  /  z ]_ z  =  Y )
65breq2d 4407 . . . . . 6  |-  ( Y  e.  V  ->  ( X R [_ Y  / 
z ]_ z  <->  X R Y ) )
74, 6bitrd 261 . . . . 5  |-  ( Y  e.  V  ->  ( [. Y  /  z ]. X R z  <->  X R Y ) )
81, 7ax-mp 5 . . . 4  |-  ( [. Y  /  z ]. X R z  <->  X R Y )
9 sbcel1v 3314 . . . 4  |-  ( [. Y  /  z ]. z  e.  A  <->  Y  e.  A
)
103, 8, 93imtr3g 277 . . 3  |-  ( [. Y  /  z ]. ( X R z  ->  z  e.  A )  ->  ( X R Y  ->  Y  e.  A ) )
112, 10syl 17 . 2  |-  ( A. z ( X R z  ->  z  e.  A )  ->  ( X R Y  ->  Y  e.  A ) )
12 frege72.x . . 3  |-  X  e.  U
1312frege71 36601 . 2  |-  ( ( A. z ( X R z  ->  z  e.  A )  ->  ( X R Y  ->  Y  e.  A ) )  -> 
( R hereditary  A  ->  ( X  e.  A  ->  ( X R Y  ->  Y  e.  A )
) ) )
1411, 13ax-mp 5 1  |-  ( R hereditary  A  ->  ( X  e.  A  ->  ( X R Y  ->  Y  e.  A ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189   A.wal 1450    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:  frege73  36603  frege74  36604
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