Theorem bnj609 30987

Description: Technical lemma for bnj852 30991. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj609.1	|- ( ph <-> ( f ` (/) ) = pred ( X , A , R ) )
bnj609.2	|- ( ph" <-> [. G / f ]. ph )
bnj609.3	|- G e. _V

Assertion

Ref	Expression
bnj609	|- ( ph" <-> ( G ` (/) ) = pred ( X , A , R ) )

Distinct variable groups:   A, f   R, f   f, X

Allowed substitution hints:   ph( f)   G( f)   ph"( f)

Proof of Theorem bnj609

Dummy variable e is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj609.2	. 2 |- ( ph" <-> [. G / f ]. ph )
2		bnj609.3	. . 3 |- G e. _V
3		dfsbcq 3437	. . 3 |- ( e = G -> ( [. e / f ]. ph <-> [. G / f ]. ph ) )
4		fveq1 6190	. . . 4 |- ( e = G -> ( e ` (/) ) = ( G ` (/) ) )
5	4	eqeq1d 2624	. . 3 |- ( e = G -> ( ( e ` (/) ) = pred ( X , A , R ) <-> ( G ` (/) ) = pred ( X , A , R ) ) )
6		bnj609.1	. . . . 5 |- ( ph <-> ( f ` (/) ) = pred ( X , A , R ) )
7	6	sbcbii 3491	. . . 4 |- ( [. e / f ]. ph <-> [. e / f ]. ( f ` (/) ) = pred ( X , A , R ) )
8		vex 3203	. . . . 5 |- e e. _V
9		fveq1 6190	. . . . . 6 |- ( f = e -> ( f ` (/) ) = ( e ` (/) ) )
10	9	eqeq1d 2624	. . . . 5 |- ( f = e -> ( ( f ` (/) ) = pred ( X , A , R ) <-> ( e ` (/) ) = pred ( X , A , R ) ) )
11	8, 10	sbcie 3470	. . . 4 |- ( [. e / f ]. ( f ` (/) ) = pred ( X , A , R ) <-> ( e ` (/) ) = pred ( X , A , R ) )
12	7, 11	bitri 264	. . 3 |- ( [. e / f ]. ph <-> ( e ` (/) ) = pred ( X , A , R ) )
13	2, 3, 5, 12	vtoclb 3263	. 2 |- ( [. G / f ]. ph <-> ( G ` (/) ) = pred ( X , A , R ) )
14	1, 13	bitri 264	1 |- ( ph" <-> ( G ` (/) ) = pred ( X , A , R ) )

Syntax hints:   <-> wb 196   = wceq 1483   e. wcel 1990   _Vcvv 3200   [.wsbc 3435   (/)c0 3915   ` cfv 5888   predc-bnj14 30754

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-v 3202  df-sbc 3436  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896

This theorem is referenced by:  bnj600  30989  bnj908  31001  bnj934  31005