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Theorem pm4.71r 615
Description: Implication in terms of biconditional and conjunction. Theorem *4.71 of [WhiteheadRussell] p. 120 (with conjunct reversed). (Contributed by NM, 25-Jul-1999.)
Assertion
Ref Expression
pm4.71r  |-  ( (
ph  ->  ps )  <->  ( ph  <->  ( ps  /\  ph )
) )

Proof of Theorem pm4.71r
StepHypRef Expression
1 pm4.71 614 . 2  |-  ( (
ph  ->  ps )  <->  ( ph  <->  (
ph  /\  ps )
) )
2 ancom 439 . . 3  |-  ( (
ph  /\  ps )  <->  ( ps  /\  ph )
)
32bibi2i 306 . 2  |-  ( (
ph 
<->  ( ph  /\  ps ) )  <->  ( ph  <->  ( ps  /\  ph )
) )
41, 3bitri 242 1  |-  ( (
ph  ->  ps )  <->  ( ph  <->  ( ps  /\  ph )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360
This theorem is referenced by:  pm4.71ri  617  pm4.71rd  619
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10
This theorem depends on definitions:  df-bi 179  df-an 362
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