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Metamath Proof Explorer

Theorem bnj581

Description: Technical lemma for bnj580 . 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) Remove unnecessary distinct variable conditions. (Revised by Andrew Salmon, 9-Jul-2011) (New usage is discouraged.)

Ref Expression
Hypotheses bnj581.3 
|- ( ch <-> ( f Fn n /\ ph /\ ps ) )
bnj581.4 
|- ( ph' <-> [. g / f ]. ph )
bnj581.5 
|- ( ps' <-> [. g / f ]. ps )
bnj581.6 
|- ( ch' <-> [. g / f ]. ch )
Assertion bnj581 
|- ( ch' <-> ( g Fn n /\ ph' /\ ps' ) )

Proof

Step Hyp Ref Expression
1 bnj581.3 
 |-  ( ch <-> ( f Fn n /\ ph /\ ps ) )
2 bnj581.4 
 |-  ( ph' <-> [. g / f ]. ph )
3 bnj581.5 
 |-  ( ps' <-> [. g / f ]. ps )
4 bnj581.6 
 |-  ( ch' <-> [. g / f ]. ch )
5 1 sbcbii 
 |-  ( [. g / f ]. ch <-> [. g / f ]. ( f Fn n /\ ph /\ ps ) )
6 sbc3an 
 |-  ( [. g / f ]. ( f Fn n /\ ph /\ ps ) <-> ( [. g / f ]. f Fn n /\ [. g / f ]. ph /\ [. g / f ]. ps ) )
7 bnj62 
 |-  ( [. g / f ]. f Fn n <-> g Fn n )
8 7 bicomi 
 |-  ( g Fn n <-> [. g / f ]. f Fn n )
9 8 2 3 3anbi123i 
 |-  ( ( g Fn n /\ ph' /\ ps' ) <-> ( [. g / f ]. f Fn n /\ [. g / f ]. ph /\ [. g / f ]. ps ) )
10 6 9 bitr4i 
 |-  ( [. g / f ]. ( f Fn n /\ ph /\ ps ) <-> ( g Fn n /\ ph' /\ ps' ) )
11 4 5 10 3bitri 
 |-  ( ch' <-> ( g Fn n /\ ph' /\ ps' ) )