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Theorem pm4.71 381
Description: Implication in terms of biconditional and conjunction. Theorem *4.71 of [WhiteheadRussell] p. 120. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 2-Dec-2012.)
Assertion
Ref Expression
pm4.71  |-  ( (
ph  ->  ps )  <->  ( ph  <->  (
ph  /\  ps )
) )

Proof of Theorem pm4.71
StepHypRef Expression
1 simpl 107 . . 3  |-  ( (
ph  /\  ps )  ->  ph )
21biantru 296 . 2  |-  ( (
ph  ->  ( ph  /\  ps ) )  <->  ( ( ph  ->  ( ph  /\  ps ) )  /\  (
( ph  /\  ps )  ->  ph ) ) )
3 anclb 312 . 2  |-  ( (
ph  ->  ps )  <->  ( ph  ->  ( ph  /\  ps ) ) )
4 dfbi2 380 . 2  |-  ( (
ph 
<->  ( ph  /\  ps ) )  <->  ( ( ph  ->  ( ph  /\  ps ) )  /\  (
( ph  /\  ps )  ->  ph ) ) )
52, 3, 43bitr4i 210 1  |-  ( (
ph  ->  ps )  <->  ( ph  <->  (
ph  /\  ps )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  pm4.71r  382  pm4.71i  383  pm4.71d  385  bigolden  899  pm5.75  906  exintrbi  1567  rabid2  2539  dfss2  3003  disj3  3323  dmopab3  4615
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