Theorem bnj154 30948

Description: Technical lemma for bnj153 30950. 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
bnj154.1	|- ( ph1 <-> [. g / f ]. ph' )
bnj154.2	|- ( ph' <-> ( f ` (/) ) = pred ( x , A , R ) )

Assertion

Ref	Expression
bnj154	|- ( ph1 <-> ( g ` (/) ) = pred ( x , A , R ) )

Distinct variable groups:   A, f   R, f   f, g   x, f

Allowed substitution hints:   A( x, g)   R( x, g)   ph'( x, f, g)   ph1( x, f, g)

Proof of Theorem bnj154

Step	Hyp	Ref	Expression
1		bnj154.1	. 2 |- ( ph1 <-> [. g / f ]. ph' )
2		bnj154.2	. . 3 |- ( ph' <-> ( f ` (/) ) = pred ( x , A , R ) )
3	2	sbcbii 3491	. 2 |- ( [. g / f ]. ph' <-> [. g / f ]. ( f ` (/) ) = pred ( x , A , R ) )
4		vex 3203	. . 3 |- g e. _V
5		fveq1 6190	. . . 4 |- ( f = g -> ( f ` (/) ) = ( g ` (/) ) )
6	5	eqeq1d 2624	. . 3 |- ( f = g -> ( ( f ` (/) ) = pred ( x , A , R ) <-> ( g ` (/) ) = pred ( x , A , R ) ) )
7	4, 6	sbcie 3470	. 2 |- ( [. g / f ]. ( f ` (/) ) = pred ( x , A , R ) <-> ( g ` (/) ) = pred ( x , A , R ) )
8	1, 3, 7	3bitri 286	1 |- ( ph1 <-> ( g ` (/) ) = pred ( x , A , R ) )

Syntax hints:   <-> wb 196   = wceq 1483   [.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:  bnj153  30950  bnj580  30983  bnj607  30986