Theorem frege73 38230

Description: Lemma for frege87 38244. Proposition 73 of [Frege1879] p. 59. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
frege73.x	|- X e. U
frege73.y	|- Y e. V

Assertion

Ref	Expression
frege73	|- ( ( R hereditary A -> X e. A ) -> ( R hereditary A -> ( X R Y -> Y e. A ) ) )

Proof of Theorem frege73

Step	Hyp	Ref	Expression
1		frege73.x	. . 3 |- X e. U
2		frege73.y	. . 3 |- Y e. V
3	1, 2	frege72 38229	. 2 |- ( R hereditary A -> ( X e. A -> ( X R Y -> Y e. A ) ) )
4		ax-frege2 38085	. 2 |- ( ( R hereditary A -> ( X e. A -> ( X R Y -> Y e. A ) ) ) -> ( ( R hereditary A -> X e. A ) -> ( R hereditary A -> ( X R Y -> Y e. A ) ) ) )
5	3, 4	ax-mp 5	1 |- ( ( R hereditary A -> X e. A ) -> ( R hereditary A -> ( X R Y -> Y e. A ) ) )

Syntax hints:   -> wi 4   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-frege1 38084  ax-frege2 38085  ax-frege8 38103  ax-frege52a 38151  ax-frege58b 38195

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-sbc 3436  df-csb 3534  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:  frege87  38244