Theorem 3anbi2d 1358

Description: Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.)

Hypothesis

Ref	Expression
3anbi1d.1	|- ( ph -> ( ps <-> ch ) )

Assertion

Ref	Expression
3anbi2d	|- ( ph -> ( ( th /\ ps /\ ta ) <-> ( th /\ ch /\ ta ) ) )

Proof of Theorem 3anbi2d

Step	Hyp	Ref	Expression
1		biidd 172	. 2 |- ( ph -> ( th <-> th ) )
2		3anbi1d.1	. 2 |- ( ph -> ( ps <-> ch ) )
3	1, 2	3anbi12d 1354	1 |- ( ph -> ( ( th /\ ps /\ ta ) <-> ( th /\ ch /\ ta ) ) )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   /\ w3a 1009

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This proof depends on definitions:  df-bi 117  df-3an 1011

This theorem is used by:  vtocl3gaf  2892  ordsoexmid  4709  ereq2  6815  genpelxp  7878  seq3f1olemp  10952  qexpclz  10997  mhmlem  13917  opprsubgg  14390  lmodlema  14628  ivthreinc  15746  incistruhgr  16331  issubgr2  16499