Theorem bnj887 30835

Description: /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj887.1	|- ( ph <-> ph' )
bnj887.2	|- ( ps <-> ps' )
bnj887.3	|- ( ch <-> ch' )
bnj887.4	|- ( th <-> th' )

Assertion

Ref	Expression
bnj887	|- ( ( ph /\ ps /\ ch /\ th ) <-> ( ph' /\ ps' /\ ch' /\ th' ) )

Proof of Theorem bnj887

Step	Hyp	Ref	Expression
1		bnj887.1	. . . 4 |- ( ph <-> ph' )
2		bnj887.2	. . . 4 |- ( ps <-> ps' )
3		bnj887.3	. . . 4 |- ( ch <-> ch' )
4	1, 2, 3	3anbi123i 1251	. . 3 |- ( ( ph /\ ps /\ ch ) <-> ( ph' /\ ps' /\ ch' ) )
5		bnj887.4	. . 3 |- ( th <-> th' )
6	4, 5	anbi12i 733	. 2 |- ( ( ( ph /\ ps /\ ch ) /\ th ) <-> ( ( ph' /\ ps' /\ ch' ) /\ th' ) )
7		df-bnj17 30753	. 2 |- ( ( ph /\ ps /\ ch /\ th ) <-> ( ( ph /\ ps /\ ch ) /\ th ) )
8		df-bnj17 30753	. 2 |- ( ( ph' /\ ps' /\ ch' /\ th' ) <-> ( ( ph' /\ ps' /\ ch' ) /\ th' ) )
9	6, 7, 8	3bitr4i 292	1 |- ( ( ph /\ ps /\ ch /\ th ) <-> ( ph' /\ ps' /\ ch' /\ th' ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   /\ w-bnj17 30752

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

This theorem depends on definitions:  df-bi 197  df-an 386  df-3an 1039  df-bnj17 30753

This theorem is referenced by:  bnj1040  31040  bnj1128  31058