Theorem mpbir3an 1163

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 1159	. 2 |- ( ps /\ ch /\ th )
5		mpbir3an.4	. 2 |- ( ph <-> ( ps /\ ch /\ th ) )
6	4, 5	mpbir 145	1 |- ph

Syntax hints:   <-> wb 104   /\ w3a 962

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

This theorem depends on definitions:  df-bi 116  df-3an 964

This theorem is referenced by:  limon  4429  limom  4527  issmo  6185  xpider  6500  1eluzge0  9369  2eluzge1  9371  0elunit  9769  1elunit  9770  4fvwrd4  9917  fzo0to42pr  9997  resqrexlemga  10795  sincos1sgn  11471  sincos2sgn  11472  qnnen  11944  strleun  12048  sinhalfpilem  12872  sincos4thpi  12921  sincos6thpi  12923  pigt3  12925