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Metamath Proof Explorer

Theorem bnj1040

Description: Technical lemma for bnj69 . 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 bnj1040.1 No typesetting found for |- ( ph' <-> [. j / i ]. ph ) with typecode |-
bnj1040.2 No typesetting found for |- ( ps' <-> [. j / i ]. ps ) with typecode |-
bnj1040.3 ⊢ χ ↔ n ∈ D ∧ f Fn n ∧ φ ∧ ψ
bnj1040.4 No typesetting found for |- ( ch' <-> [. j / i ]. ch ) with typecode |-
Assertion bnj1040 Could not format assertion : No typesetting found for |- ( ch' <-> ( n e. D /\ f Fn n /\ ph' /\ ps' ) ) with typecode |-

Proof

Step Hyp Ref Expression
1 bnj1040.1 Could not format ( ph' <-> [. j / i ]. ph ) : No typesetting found for |- ( ph' <-> [. j / i ]. ph ) with typecode |-
2 bnj1040.2 Could not format ( ps' <-> [. j / i ]. ps ) : No typesetting found for |- ( ps' <-> [. j / i ]. ps ) with typecode |-
3 bnj1040.3 ⊢ χ ↔ n ∈ D ∧ f Fn n ∧ φ ∧ ψ
4 bnj1040.4 Could not format ( ch' <-> [. j / i ]. ch ) : No typesetting found for |- ( ch' <-> [. j / i ]. ch ) with typecode |-
5 3 sbcbii ⊢ [˙j / i]˙ χ ↔ [˙j / i]˙ n ∈ D ∧ f Fn n ∧ φ ∧ ψ
6 df-bnj17 ⊢ [˙j / i]˙ n ∈ D ∧ [˙j / i]˙ f Fn n ∧ [˙j / i]˙ φ ∧ [˙j / i]˙ ψ ↔ [˙j / i]˙ n ∈ D ∧ [˙j / i]˙ f Fn n ∧ [˙j / i]˙ φ ∧ [˙j / i]˙ ψ
7 vex ⊢ j ∈ V
8 7 bnj525 ⊢ [˙j / i]˙ n ∈ D ↔ n ∈ D
9 8 bicomi ⊢ n ∈ D ↔ [˙j / i]˙ n ∈ D
10 7 bnj525 ⊢ [˙j / i]˙ f Fn n ↔ f Fn n
11 10 bicomi ⊢ f Fn n ↔ [˙j / i]˙ f Fn n
12 9 11 1 2 bnj887 Could not format ( ( n e. D /\ f Fn n /\ ph' /\ ps' ) <-> ( [. j / i ]. n e. D /\ [. j / i ]. f Fn n /\ [. j / i ]. ph /\ [. j / i ]. ps ) ) : No typesetting found for |- ( ( n e. D /\ f Fn n /\ ph' /\ ps' ) <-> ( [. j / i ]. n e. D /\ [. j / i ]. f Fn n /\ [. j / i ]. ph /\ [. j / i ]. ps ) ) with typecode |-
13 df-bnj17 ⊢ n ∈ D ∧ f Fn n ∧ φ ∧ ψ ↔ n ∈ D ∧ f Fn n ∧ φ ∧ ψ
14 13 sbcbii ⊢ [˙j / i]˙ n ∈ D ∧ f Fn n ∧ φ ∧ ψ ↔ [˙j / i]˙ n ∈ D ∧ f Fn n ∧ φ ∧ ψ
15 sbcan ⊢ [˙j / i]˙ n ∈ D ∧ f Fn n ∧ φ ∧ ψ ↔ [˙j / i]˙ n ∈ D ∧ f Fn n ∧ φ ∧ [˙j / i]˙ ψ
16 sbc3an ⊢ [˙j / i]˙ n ∈ D ∧ f Fn n ∧ φ ↔ [˙j / i]˙ n ∈ D ∧ [˙j / i]˙ f Fn n ∧ [˙j / i]˙ φ
17 16 anbi1i ⊢ [˙j / i]˙ n ∈ D ∧ f Fn n ∧ φ ∧ [˙j / i]˙ ψ ↔ [˙j / i]˙ n ∈ D ∧ [˙j / i]˙ f Fn n ∧ [˙j / i]˙ φ ∧ [˙j / i]˙ ψ
18 14 15 17 3bitri ⊢ [˙j / i]˙ n ∈ D ∧ f Fn n ∧ φ ∧ ψ ↔ [˙j / i]˙ n ∈ D ∧ [˙j / i]˙ f Fn n ∧ [˙j / i]˙ φ ∧ [˙j / i]˙ ψ
19 6 12 18 3bitr4ri Could not format ( [. j / i ]. ( n e. D /\ f Fn n /\ ph /\ ps ) <-> ( n e. D /\ f Fn n /\ ph' /\ ps' ) ) : No typesetting found for |- ( [. j / i ]. ( n e. D /\ f Fn n /\ ph /\ ps ) <-> ( n e. D /\ f Fn n /\ ph' /\ ps' ) ) with typecode |-
20 4 5 19 3bitri Could not format ( ch' <-> ( n e. D /\ f Fn n /\ ph' /\ ps' ) ) : No typesetting found for |- ( ch' <-> ( n e. D /\ f Fn n /\ ph' /\ ps' ) ) with typecode |-