Theorem frege116 38273

Description: One direction of dffrege115 38272. Proposition 116 of [Frege1879] p. 77. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
frege116.x	|- X e. U

Assertion

Ref	Expression
frege116	|- ( Fun `' `' R -> A. b ( b R X -> A. a ( b R a -> a = X ) ) )

Distinct variable groups:   a, b, R   X, a, b

Allowed substitution hints:   U( a, b)

Proof of Theorem frege116

Dummy variable c is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dffrege115 38272	. 2 |- ( A. c A. b ( b R c -> A. a ( b R a -> a = c ) ) <-> Fun `' `' R )
2		frege116.x	. . . 4 |- X e. U
3	2	frege68c 38225	. . 3 |- ( ( A. c A. b ( b R c -> A. a ( b R a -> a = c ) ) <-> Fun `' `' R ) -> ( Fun `' `' R -> [. X / c ]. A. b ( b R c -> A. a ( b R a -> a = c ) ) ) )
4		sbcal 3485	. . . 4 |- ( [. X / c ]. A. b ( b R c -> A. a ( b R a -> a = c ) ) <-> A. b [. X / c ]. ( b R c -> A. a ( b R a -> a = c ) ) )
5		sbcimg 3477	. . . . . . 7 |- ( X e. U -> ( [. X / c ]. ( b R c -> A. a ( b R a -> a = c ) ) <-> ( [. X / c ]. b R c -> [. X / c ]. A. a ( b R a -> a = c ) ) ) )
6	2, 5	ax-mp 5	. . . . . 6 |- ( [. X / c ]. ( b R c -> A. a ( b R a -> a = c ) ) <-> ( [. X / c ]. b R c -> [. X / c ]. A. a ( b R a -> a = c ) ) )
7		sbcbr2g 4710	. . . . . . . . 9 |- ( X e. U -> ( [. X / c ]. b R c <-> b R [_ X / c ]_ c ) )
8	2, 7	ax-mp 5	. . . . . . . 8 |- ( [. X / c ]. b R c <-> b R [_ X / c ]_ c )
9		csbvarg 4003	. . . . . . . . . 10 |- ( X e. U -> [_ X / c ]_ c = X )
10	2, 9	ax-mp 5	. . . . . . . . 9 |- [_ X / c ]_ c = X
11	10	breq2i 4661	. . . . . . . 8 |- ( b R [_ X / c ]_ c <-> b R X )
12	8, 11	bitri 264	. . . . . . 7 |- ( [. X / c ]. b R c <-> b R X )
13		sbcal 3485	. . . . . . . 8 |- ( [. X / c ]. A. a ( b R a -> a = c ) <-> A. a [. X / c ]. ( b R a -> a = c ) )
14		sbcimg 3477	. . . . . . . . . . 11 |- ( X e. U -> ( [. X / c ]. ( b R a -> a = c ) <-> ( [. X / c ]. b R a -> [. X / c ]. a = c ) ) )
15	2, 14	ax-mp 5	. . . . . . . . . 10 |- ( [. X / c ]. ( b R a -> a = c ) <-> ( [. X / c ]. b R a -> [. X / c ]. a = c ) )
16		sbcg 3503	. . . . . . . . . . . 12 |- ( X e. U -> ( [. X / c ]. b R a <-> b R a ) )
17	2, 16	ax-mp 5	. . . . . . . . . . 11 |- ( [. X / c ]. b R a <-> b R a )
18		sbceq2g 3990	. . . . . . . . . . . . 13 |- ( X e. U -> ( [. X / c ]. a = c <-> a = [_ X / c ]_ c ) )
19	2, 18	ax-mp 5	. . . . . . . . . . . 12 |- ( [. X / c ]. a = c <-> a = [_ X / c ]_ c )
20	10	eqeq2i 2634	. . . . . . . . . . . 12 |- ( a = [_ X / c ]_ c <-> a = X )
21	19, 20	bitri 264	. . . . . . . . . . 11 |- ( [. X / c ]. a = c <-> a = X )
22	17, 21	imbi12i 340	. . . . . . . . . 10 |- ( ( [. X / c ]. b R a -> [. X / c ]. a = c ) <-> ( b R a -> a = X ) )
23	15, 22	bitri 264	. . . . . . . . 9 |- ( [. X / c ]. ( b R a -> a = c ) <-> ( b R a -> a = X ) )
24	23	albii 1747	. . . . . . . 8 |- ( A. a [. X / c ]. ( b R a -> a = c ) <-> A. a ( b R a -> a = X ) )
25	13, 24	bitri 264	. . . . . . 7 |- ( [. X / c ]. A. a ( b R a -> a = c ) <-> A. a ( b R a -> a = X ) )
26	12, 25	imbi12i 340	. . . . . 6 |- ( ( [. X / c ]. b R c -> [. X / c ]. A. a ( b R a -> a = c ) ) <-> ( b R X -> A. a ( b R a -> a = X ) ) )
27	6, 26	bitri 264	. . . . 5 |- ( [. X / c ]. ( b R c -> A. a ( b R a -> a = c ) ) <-> ( b R X -> A. a ( b R a -> a = X ) ) )
28	27	albii 1747	. . . 4 |- ( A. b [. X / c ]. ( b R c -> A. a ( b R a -> a = c ) ) <-> A. b ( b R X -> A. a ( b R a -> a = X ) ) )
29	4, 28	bitri 264	. . 3 |- ( [. X / c ]. A. b ( b R c -> A. a ( b R a -> a = c ) ) <-> A. b ( b R X -> A. a ( b R a -> a = X ) ) )
30	3, 29	syl6ib 241	. 2 |- ( ( A. c A. b ( b R c -> A. a ( b R a -> a = c ) ) <-> Fun `' `' R ) -> ( Fun `' `' R -> A. b ( b R X -> A. a ( b R a -> a = X ) ) ) )
31	1, 30	ax-mp 5	1 |- ( Fun `' `' R -> A. b ( b R X -> A. a ( b R a -> a = X ) ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   = wceq 1483   e. wcel 1990   [.wsbc 3435   [_csb 3533   class class class wbr 4653   `'ccnv 5113   Fun wfun 5882

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-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-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-fun 5890

This theorem is referenced by:  frege117  38274