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Metamath Proof Explorer

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 ∈ V
5 fveq1 ⊢ f = g → f ⁡ ∅ = g ⁡ ∅
6 5 eqeq1d ⊢ f = g → f ⁡ ∅ = pred x A R ↔ g ⁡ ∅ = pred x A R
7 4 6 sbcie ⊢ [˙g / f]˙ f ⁡ ∅ = pred x A R ↔ g ⁡ ∅ = pred x A R
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 |-