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Metamath Proof Explorer

Theorem bnj609

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 bnj609.1 ⊢ φ ↔ f ⁡ ∅ = pred X A R
bnj609.2 No typesetting found for |- ( ph" <-> [. G / f ]. ph ) with typecode |-
bnj609.3 ⊢ G ∈ V
Assertion bnj609 Could not format assertion : No typesetting found for |- ( ph" <-> ( G ` (/) ) = _pred ( X , A , R ) ) with typecode |-

Proof

Step Hyp Ref Expression
1 bnj609.1 ⊢ φ ↔ f ⁡ ∅ = pred X A R
2 bnj609.2 Could not format ( ph" <-> [. G / f ]. ph ) : No typesetting found for |- ( ph" <-> [. G / f ]. ph ) with typecode |-
3 bnj609.3 ⊢ G ∈ V
4 dfsbcq ⊢ e = G → [˙e / f]˙ φ ↔ [˙G / f]˙ φ
5 fveq1 ⊢ e = G → e ⁡ ∅ = G ⁡ ∅
6 5 eqeq1d ⊢ e = G → e ⁡ ∅ = pred X A R ↔ G ⁡ ∅ = pred X A R
7 1 sbcbii ⊢ [˙e / f]˙ φ ↔ [˙e / f]˙ f ⁡ ∅ = pred X A R
8 vex ⊢ e ∈ V
9 fveq1 ⊢ f = e → f ⁡ ∅ = e ⁡ ∅
10 9 eqeq1d ⊢ f = e → f ⁡ ∅ = pred X A R ↔ e ⁡ ∅ = pred X A R
11 8 10 sbcie ⊢ [˙e / f]˙ f ⁡ ∅ = pred X A R ↔ e ⁡ ∅ = pred X A R
12 7 11 bitri ⊢ [˙e / f]˙ φ ↔ e ⁡ ∅ = pred X A R
13 3 4 6 12 vtoclb ⊢ [˙G / f]˙ φ ↔ G ⁡ ∅ = pred X A R
14 2 13 bitri Could not format ( ph" <-> ( G ` (/) ) = _pred ( X , A , R ) ) : No typesetting found for |- ( ph" <-> ( G ` (/) ) = _pred ( X , A , R ) ) with typecode |-