Theorem 19.26-3an 1800

Description: Theorem 19.26 1798 with triple conjunction. (Contributed by NM, 13-Sep-2011.)

Assertion

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

Proof of Theorem 19.26-3an

Step	Hyp	Ref	Expression
1		19.26 1798	. . 3 |- ( A. x ( ( ph /\ ps ) /\ ch ) <-> ( A. x ( ph /\ ps ) /\ A. x ch ) )
2		19.26 1798	. . . 4 |- ( A. x ( ph /\ ps ) <-> ( A. x ph /\ A. x ps ) )
3	2	anbi1i 731	. . 3 |- ( ( A. x ( ph /\ ps ) /\ A. x ch ) <-> ( ( A. x ph /\ A. x ps ) /\ A. x ch ) )
4	1, 3	bitri 264	. 2 |- ( A. x ( ( ph /\ ps ) /\ ch ) <-> ( ( A. x ph /\ A. x ps ) /\ A. x ch ) )
5		df-3an 1039	. . 3 |- ( ( ph /\ ps /\ ch ) <-> ( ( ph /\ ps ) /\ ch ) )
6	5	albii 1747	. 2 |- ( A. x ( ph /\ ps /\ ch ) <-> A. x ( ( ph /\ ps ) /\ ch ) )
7		df-3an 1039	. 2 |- ( ( A. x ph /\ A. x ps /\ A. x ch ) <-> ( ( A. x ph /\ A. x ps ) /\ A. x ch ) )
8	4, 6, 7	3bitr4i 292	1 |- ( A. x ( ph /\ ps /\ ch ) <-> ( A. x ph /\ A. x ps /\ A. x ch ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   A.wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737

This theorem depends on definitions:  df-bi 197  df-an 386  df-3an 1039

This theorem is referenced by:  alrim3con13v  38743  19.21a3con13vVD  39087