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Theorem pm4.71 611
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 443 . . 3  |-  ( (
ph  /\  ps )  ->  ph )
21biantru 491 . 2  |-  ( (
ph  ->  ( ph  /\  ps ) )  <->  ( ( ph  ->  ( ph  /\  ps ) )  /\  (
( ph  /\  ps )  ->  ph ) ) )
3 anclb 530 . 2  |-  ( (
ph  ->  ps )  <->  ( ph  ->  ( ph  /\  ps ) ) )
4 dfbi2 609 . 2  |-  ( (
ph 
<->  ( ph  /\  ps ) )  <->  ( ( ph  ->  ( ph  /\  ps ) )  /\  (
( ph  /\  ps )  ->  ph ) ) )
52, 3, 43bitr4i 268 1  |-  ( (
ph  ->  ps )  <->  ( ph  <->  (
ph  /\  ps )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358
This theorem is referenced by:  pm4.71r  612  pm4.71i  613  pm4.71d  615  bigolden  901  pm5.75  903  exintrbi  1603  rabid2  2730  dfss2  3182  disj3  3512  dmopab3  4907  rabid2f  23151  mptfnf  23241
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 177  df-an 360
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