Theorem ccased 758

Description: Deduction for combining cases.

Hypotheses

Ref	Expression
ccased.1	|- (ph -> ((ps /\ ch) -> et))
ccased.2	|- (ph -> ((th /\ ch) -> et))
ccased.3	|- (ph -> ((ps /\ ta) -> et))
ccased.4	|- (ph -> ((th /\ ta) -> et))

Assertion

Ref	Expression
ccased	|- (ph -> (((ps \/ th) /\ (ch \/ ta)) -> et))

Proof of Theorem ccased

Step	Hyp	Ref	Expression
1		ccased.1	. . . 4 |- (ph -> ((ps /\ ch) -> et))
2		ccased.2	. . . 4 |- (ph -> ((th /\ ch) -> et))
3	1, 2	jaod 426	. . 3 |- (ph -> (((ps /\ ch) \/ (th /\ ch)) -> et))
4		ccased.3	. . . 4 |- (ph -> ((ps /\ ta) -> et))
5		ccased.4	. . . 4 |- (ph -> ((th /\ ta) -> et))
6	4, 5	jaod 426	. . 3 |- (ph -> (((ps /\ ta) \/ (th /\ ta)) -> et))
7	3, 6	jaod 426	. 2 |- (ph -> ((((ps /\ ch) \/ (th /\ ch)) \/ ((ps /\ ta) \/ (th /\ ta))) -> et))
8		anddi 609	. . 3 |- (((ps \/ th) /\ (ch \/ ta)) <-> (((ps /\ ch) \/ (ps /\ ta)) \/ ((th /\ ch) \/ (th /\ ta))))
9		or4 264	. . 3 |- ((((ps /\ ch) \/ (ps /\ ta)) \/ ((th /\ ch) \/ (th /\ ta))) <-> (((ps /\ ch) \/ (th /\ ch)) \/ ((ps /\ ta) \/ (th /\ ta))))
10	8, 9	bitr 173	. 2 |- (((ps \/ th) /\ (ch \/ ta)) <-> (((ps /\ ch) \/ (th /\ ch)) \/ ((ps /\ ta) \/ (th /\ ta))))
11	7, 10	syl5ib 206	1 |- (ph -> (((ps \/ th) /\ (ch \/ ta)) -> et))

Syntax hints:   -> wi 3   \/ wo 222   /\ wa 223

This theorem is referenced by:  zaddclt 6167  zmulclt 6182  zltp1let 6183

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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