Theorem mpbir3an 1131

Description: Detach a conjunction of truths in a biconditional. (Contributed by NM, 16-Sep-2011.) (Revised by NM, 9-Jan-2015.)

Hypotheses

Ref	Expression
mpbir3an.1	|- ps
mpbir3an.2	|- ch
mpbir3an.3	|- th
mpbir3an.4	|- ( ph <-> ( ps /\ ch /\ th ) )

Assertion

Ref	Expression
mpbir3an	|- ph

Proof of Theorem mpbir3an

Step	Hyp	Ref	Expression
1		mpbir3an.1	. . 3 |- ps
2		mpbir3an.2	. . 3 |- ch
3		mpbir3an.3	. . 3 |- th
4	1, 2, 3	3pm3.2i 1127	. 2 |- ( ps /\ ch /\ th )
5		mpbir3an.4	. 2 |- ( ph <-> ( ps /\ ch /\ th ) )
6	4, 5	mpbir 145	1 |- ph

Syntax hints:   <-> wb 104   /\ w3a 930

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116  df-3an 932

This theorem is referenced by:  limon  4367  limom  4465  issmo  6115  xpider  6430  1eluzge0  9219  2eluzge1  9221  0elunit  9610  1elunit  9611  4fvwrd4  9758  fzo0to42pr  9838  resqrexlemga  10635  sincos1sgn  11269  sincos2sgn  11270  qnnen  11736  strleun  11830