Theorem jcab 596

Description: Distributive law for implication over conjunction. Compare Theorem *4.76 of [WhiteheadRussell] p. 121.

Assertion

Ref	Expression
jcab	|- ((ph -> (ps /\ ch)) <-> ((ph -> ps) /\ (ph -> ch)))

Proof of Theorem jcab

Step	Hyp	Ref	Expression
1		ordi 594	. 2 |- ((-. ph \/ (ps /\ ch)) <-> ((-. ph \/ ps) /\ (-. ph \/ ch)))
2		imor 234	. 2 |- ((ph -> (ps /\ ch)) <-> (-. ph \/ (ps /\ ch)))
3		imor 234	. . 3 |- ((ph -> ps) <-> (-. ph \/ ps))
4		imor 234	. . 3 |- ((ph -> ch) <-> (-. ph \/ ch))
5	3, 4	anbi12i 481	. 2 |- (((ph -> ps) /\ (ph -> ch)) <-> ((-. ph \/ ps) /\ (-. ph \/ ch)))
6	1, 2, 5	3bitr4 183	1 |- ((ph -> (ps /\ ch)) <-> ((ph -> ps) /\ (ph -> ch)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223

This theorem is referenced by:  pm4.76 597  pm3.43 601  pm5.44 685  mopick2 1429  2eu4 1445  r19.26 1742  ssconb 2160  tfr3 3911  suppsr2 5195  suppsr3 5196  pre-axsup 5263  ivthlem7 7222  ivthlem7OLD 7231

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225

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