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Metamath Proof Explorer

Theorem 19.26-3an

Description: Theorem 19.26 with triple conjunction. (Contributed by NM, 13-Sep-2011)

Ref Expression
Assertion 19.26-3an 
|- ( A. x ( ph /\ ps /\ ch ) <-> ( A. x ph /\ A. x ps /\ A. x ch ) )

Proof

Step Hyp Ref Expression
1 19.26 
 |-  ( A. x ( ph /\ ps ) <-> ( A. x ph /\ A. x ps ) )
2 1 anbi1i 
 |-  ( ( A. x ( ph /\ ps ) /\ A. x ch ) <-> ( ( A. x ph /\ A. x ps ) /\ A. x ch ) )
3 df-3an 
 |-  ( ( ph /\ ps /\ ch ) <-> ( ( ph /\ ps ) /\ ch ) )
4 3 albii 
 |-  ( A. x ( ph /\ ps /\ ch ) <-> A. x ( ( ph /\ ps ) /\ ch ) )
5 19.26 
 |-  ( A. x ( ( ph /\ ps ) /\ ch ) <-> ( A. x ( ph /\ ps ) /\ A. x ch ) )
6 4 5 bitri 
 |-  ( A. x ( ph /\ ps /\ ch ) <-> ( A. x ( ph /\ ps ) /\ A. x ch ) )
7 df-3an 
 |-  ( ( A. x ph /\ A. x ps /\ A. x ch ) <-> ( ( A. x ph /\ A. x ps ) /\ A. x ch ) )
8 2 6 7 3bitr4i 
 |-  ( A. x ( ph /\ ps /\ ch ) <-> ( A. x ph /\ A. x ps /\ A. x ch ) )