# Metamath Proof Explorer

## Theorem bnj526

Description: Technical lemma for bnj852 . 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 bnj526.1 ${⊢}{\phi }↔{f}\left(\varnothing \right)=pred\left({X},{A},{R}\right)$
bnj526.2 No typesetting found for |- ( ph" <-> [. G / f ]. ph ) with typecode |-
bnj526.3 ${⊢}{G}\in \mathrm{V}$
Assertion bnj526 Could not format assertion : No typesetting found for |- ( ph" <-> ( G  (/) ) = _pred ( X , A , R ) ) with typecode |-

### Proof

Step Hyp Ref Expression
1 bnj526.1 ${⊢}{\phi }↔{f}\left(\varnothing \right)=pred\left({X},{A},{R}\right)$
2 bnj526.2 Could not format ( ph" <-> [. G / f ]. ph ) : No typesetting found for |- ( ph" <-> [. G / f ]. ph ) with typecode |-
3 bnj526.3 ${⊢}{G}\in \mathrm{V}$
4 1 sbcbii
5 fveq1 ${⊢}{f}={G}\to {f}\left(\varnothing \right)={G}\left(\varnothing \right)$
6 5 eqeq1d ${⊢}{f}={G}\to \left({f}\left(\varnothing \right)=pred\left({X},{A},{R}\right)↔{G}\left(\varnothing \right)=pred\left({X},{A},{R}\right)\right)$
7 3 6 sbcie
8 2 4 7 3bitri Could not format ( ph" <-> ( G  (/) ) = _pred ( X , A , R ) ) : No typesetting found for |- ( ph" <-> ( G ` (/) ) = _pred ( X , A , R ) ) with typecode |-