Theorem bnj934 31005

Description: Technical lemma for bnj69 31078. 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
bnj934.1	|- ( ph <-> ( f ` (/) ) = pred ( X , A , R ) )
bnj934.4	|- ( ph' <-> [. p / n ]. ph )
bnj934.7	|- ( ph" <-> [. G / f ]. ph' )
bnj934.50	|- G e. _V

Assertion

Ref	Expression
bnj934	|- ( ( ph /\ ( G ` (/) ) = ( f ` (/) ) ) -> ph" )

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

Allowed substitution hints:   ph( f, n, p)   A( p)   R( p)   G( f, n, p)   X( p)   ph'( f, n, p)   ph"( f, n, p)

Proof of Theorem bnj934

Step	Hyp	Ref	Expression
1		bnj934.1	. . . 4 |- ( ph <-> ( f ` (/) ) = pred ( X , A , R ) )
2		eqtr 2641	. . . 4 |- ( ( ( G ` (/) ) = ( f ` (/) ) /\ ( f ` (/) ) = pred ( X , A , R ) ) -> ( G ` (/) ) = pred ( X , A , R ) )
3	1, 2	sylan2b 492	. . 3 |- ( ( ( G ` (/) ) = ( f ` (/) ) /\ ph ) -> ( G ` (/) ) = pred ( X , A , R ) )
4		bnj934.7	. . . . 5 |- ( ph" <-> [. G / f ]. ph' )
5		bnj934.4	. . . . . . . 8 |- ( ph' <-> [. p / n ]. ph )
6		vex 3203	. . . . . . . 8 |- p e. _V
7	1, 5, 6	bnj523 30957	. . . . . . 7 |- ( ph' <-> ( f ` (/) ) = pred ( X , A , R ) )
8	7, 1	bitr4i 267	. . . . . 6 |- ( ph' <-> ph )
9	8	sbcbii 3491	. . . . 5 |- ( [. G / f ]. ph' <-> [. G / f ]. ph )
10	4, 9	bitri 264	. . . 4 |- ( ph" <-> [. G / f ]. ph )
11		bnj934.50	. . . 4 |- G e. _V
12	1, 10, 11	bnj609 30987	. . 3 |- ( ph" <-> ( G ` (/) ) = pred ( X , A , R ) )
13	3, 12	sylibr 224	. 2 |- ( ( ( G ` (/) ) = ( f ` (/) ) /\ ph ) -> ph" )
14	13	ancoms 469	1 |- ( ( ph /\ ( G ` (/) ) = ( f ` (/) ) ) -> ph" )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = 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:  bnj929  31006