Theorem bnj154

Description: Technical lemma for bnj153 . 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.)

	Ref	Expression
Hypotheses	bnj154.1	No typesetting found for |- ( ph1 <-> [. g / f ]. ph' ) with typecode |-
	bnj154.2	No typesetting found for |- ( ph' <-> ( f ` (/) ) = _pred ( x , A , R ) ) with typecode |-
Assertion	bnj154	Could not format assertion : No typesetting found for |- ( ph1 <-> ( g ` (/) ) = _pred ( x , A , R ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		bnj154.1	Could not format ( ph1 <-> [. g / f ]. ph' ) : No typesetting found for |- ( ph1 <-> [. g / f ]. ph' ) with typecode |-
2		bnj154.2	Could not format ( ph' <-> ( f ` (/) ) = _pred ( x , A , R ) ) : No typesetting found for |- ( ph' <-> ( f ` (/) ) = _pred ( x , A , R ) ) with typecode |-
3	2	sbcbii	Could not format ( [. g / f ]. ph' <-> [. g / f ]. ( f ` (/) ) = _pred ( x , A , R ) ) : No typesetting found for |- ( [. g / f ]. ph' <-> [. g / f ]. ( f ` (/) ) = _pred ( x , A , R ) ) with typecode |-
4		vex	$$⊢g\in \mathrm{V}$$
5		fveq1	$$⊢f=g\to f\left(\varnothing \right)=g\left(\varnothing \right)$$
6	5	eqeq1d	$$⊢f=g\to \left(f\left(\varnothing \right)=pred\left(x,A,R\right)\leftrightarrow g\left(\varnothing \right)=pred\left(x,A,R\right)\right)$$
7	4 6	sbcie	$$⊢\underset{\dot{}}{[}g/f\underset{\dot{}}{]}f\left(\varnothing \right)=pred\left(x,A,R\right)\leftrightarrow g\left(\varnothing \right)=pred\left(x,A,R\right)$$
8	1 3 7	3bitri	Could not format ( ph1 <-> ( g ` (/) ) = _pred ( x , A , R ) ) : No typesetting found for |- ( ph1 <-> ( g ` (/) ) = _pred ( x , A , R ) ) with typecode |-