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Metamath Proof Explorer

Theorem moantr

Description: Sufficient condition for transitivity of conjunctions inside existential quantifiers. (Contributed by Peter Mazsa, 2-Oct-2018)

Ref Expression
Assertion moantr 
|- ( E* x ps -> ( ( E. x ( ph /\ ps ) /\ E. x ( ps /\ ch ) ) -> E. x ( ph /\ ch ) ) )

Proof

Step Hyp Ref Expression
1 exancom 
 |-  ( E. x ( ph /\ ps ) <-> E. x ( ps /\ ph ) )
2 1 anbi1i 
 |-  ( ( E. x ( ph /\ ps ) /\ E. x ( ps /\ ch ) ) <-> ( E. x ( ps /\ ph ) /\ E. x ( ps /\ ch ) ) )
3 2 anbi2i 
 |-  ( ( E* x ps /\ ( E. x ( ph /\ ps ) /\ E. x ( ps /\ ch ) ) ) <-> ( E* x ps /\ ( E. x ( ps /\ ph ) /\ E. x ( ps /\ ch ) ) ) )
4 3anass 
 |-  ( ( E* x ps /\ E. x ( ps /\ ph ) /\ E. x ( ps /\ ch ) ) <-> ( E* x ps /\ ( E. x ( ps /\ ph ) /\ E. x ( ps /\ ch ) ) ) )
5 3 4 bitr4i 
 |-  ( ( E* x ps /\ ( E. x ( ph /\ ps ) /\ E. x ( ps /\ ch ) ) ) <-> ( E* x ps /\ E. x ( ps /\ ph ) /\ E. x ( ps /\ ch ) ) )
6 mopick2 
 |-  ( ( E* x ps /\ E. x ( ps /\ ph ) /\ E. x ( ps /\ ch ) ) -> E. x ( ps /\ ph /\ ch ) )
7 5 6 sylbi 
 |-  ( ( E* x ps /\ ( E. x ( ph /\ ps ) /\ E. x ( ps /\ ch ) ) ) -> E. x ( ps /\ ph /\ ch ) )
8 3anass 
 |-  ( ( ps /\ ph /\ ch ) <-> ( ps /\ ( ph /\ ch ) ) )
9 8 exbii 
 |-  ( E. x ( ps /\ ph /\ ch ) <-> E. x ( ps /\ ( ph /\ ch ) ) )
10 exsimpr 
 |-  ( E. x ( ps /\ ( ph /\ ch ) ) -> E. x ( ph /\ ch ) )
11 9 10 sylbi 
 |-  ( E. x ( ps /\ ph /\ ch ) -> E. x ( ph /\ ch ) )
12 7 11 syl 
 |-  ( ( E* x ps /\ ( E. x ( ph /\ ps ) /\ E. x ( ps /\ ch ) ) ) -> E. x ( ph /\ ch ) )
13 impexp 
 |-  ( ( ( E* x ps /\ ( E. x ( ph /\ ps ) /\ E. x ( ps /\ ch ) ) ) -> E. x ( ph /\ ch ) ) <-> ( E* x ps -> ( ( E. x ( ph /\ ps ) /\ E. x ( ps /\ ch ) ) -> E. x ( ph /\ ch ) ) ) )
14 12 13 mpbi 
 |-  ( E* x ps -> ( ( E. x ( ph /\ ps ) /\ E. x ( ps /\ ch ) ) -> E. x ( ph /\ ch ) ) )