Theorem 3anbi13d 1355

Description: Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006.)

Hypotheses

Ref	Expression
3anbi12d.1	|- ( ph -> ( ps <-> ch ) )
3anbi12d.2	|- ( ph -> ( th <-> ta ) )

Assertion

Ref	Expression
3anbi13d	|- ( ph -> ( ( ps /\ et /\ th ) <-> ( ch /\ et /\ ta ) ) )

Proof of Theorem 3anbi13d

Step	Hyp	Ref	Expression
1		3anbi12d.1	. 2 |- ( ph -> ( ps <-> ch ) )
2		biidd 172	. 2 |- ( ph -> ( et <-> et ) )
3		3anbi12d.2	. 2 |- ( ph -> ( th <-> ta ) )
4	1, 2, 3	3anbi123d 1353	1 |- ( ph -> ( ( ps /\ et /\ th ) <-> ( ch /\ et /\ 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:  3anbi3d  1359  tfr1onlemaccex  6619  tfrcllemaccex  6632  ltxrlt  8391  opprsubgg  14390  lsspropdg  14768  islidlm  14816  lmres  15349  ivthreinc  15746  umgrvad2edg  16452