Theorem jcab 454

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 452	. 2 |- ((-. ph \/ (ps /\ ch)) <-> ((-. ph \/ ps) /\ (-. ph \/ ch)))
2		imor 204	. 2 |- ((ph -> (ps /\ ch)) <-> (-. ph \/ (ps /\ ch)))
3		imor 204	. . 3 |- ((ph -> ps) <-> (-. ph \/ ps))
4		imor 204	. . 3 |- ((ph -> ch) <-> (-. ph \/ ch))
5	3, 4	anbi12i 369	. 2 |- (((ph -> ps) /\ (ph -> ch)) <-> ((-. ph \/ ps) /\ (-. ph \/ ch)))
6	1, 2, 5	3bitr4 158	1 |- ((ph -> (ps /\ ch)) <-> ((ph -> ps) /\ (ph -> ch)))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   \/ wo 195   /\ wa 196

This theorem is referenced by:  mopick2 1057  2eu4 1070  r19.26 1289  ssconb 1598  tz7.2 2183  tfr3 2964  suppsr2 4017  suppsr3 4018  axsup 4088

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

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198

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