Theorem frege75 38232

Description: If from the proposition that x has property A, whatever x may be, it can be inferred that every result of an application of the procedure R to x has property A, then property A is hereditary in the R-sequence. Proposition 75 of [Frege1879] p. 60. (Contributed by RP, 28-Mar-2020.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege75	|- ( A. x ( x e. A -> A. y ( x R y -> y e. A ) ) -> R hereditary A )

Distinct variable groups:   x, y, A   x, R, y

Proof of Theorem frege75

Step	Hyp	Ref	Expression
1		dffrege69 38226	. 2 |- ( A. x ( x e. A -> A. y ( x R y -> y e. A ) ) <-> R hereditary A )
2		frege52aid 38152	. 2 |- ( ( A. x ( x e. A -> A. y ( x R y -> y e. A ) ) <-> R hereditary A ) -> ( A. x ( x e. A -> A. y ( x R y -> y e. A ) ) -> R hereditary A ) )
3	1, 2	ax-mp 5	1 |- ( A. x ( x e. A -> A. y ( x R y -> y e. A ) ) -> R hereditary A )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   e. wcel 1990   class class class wbr 4653   hereditary whe 38066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-frege52a 38151

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-he 38067

This theorem is referenced by:  frege97  38254  frege109  38266  frege131  38288