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Theorem pm4.71 634
Description: Implication in terms of biconditional and conjunction. Theorem *4.71 of [WhiteheadRussell] p. 120. (Contributed by NM, 21-Jun-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 458 . . 3  |-  ( (
ph  /\  ps )  ->  ph )
21biantru 507 . 2  |-  ( (
ph  ->  ( ph  /\  ps ) )  <->  ( ( ph  ->  ( ph  /\  ps ) )  /\  (
( ph  /\  ps )  ->  ph ) ) )
3 anclb 549 . 2  |-  ( (
ph  ->  ps )  <->  ( ph  ->  ( ph  /\  ps ) ) )
4 dfbi2 632 . 2  |-  ( (
ph 
<->  ( ph  /\  ps ) )  <->  ( ( ph  ->  ( ph  /\  ps ) )  /\  (
( ph  /\  ps )  ->  ph ) ) )
52, 3, 43bitr4i 280 1  |-  ( (
ph  ->  ps )  <->  ( ph  <->  (
ph  /\  ps )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 188  df-an 372
This theorem is referenced by:  pm4.71r  635  pm4.71i  636  pm4.71d  638  bigolden  943  pm5.75  945  exintrbi  1745  rabid2  3006  dfss2  3453  disj3  3837  dmopab3  5062  mptfnf  5713  cusgrauvtxb  25209  rabid2f  28122  nanorxor  36510
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