Theorem intn3an1d 10428

Description: Introduction of a conjunct inside a contradiction.

Hypothesis

Ref	Expression
intn3and.1	|- (ph -> -. ps)

Assertion

Ref	Expression
intn3an1d	|- (ph -> -. (ps /\ ch /\ th))

Proof of Theorem intn3an1d

Step	Hyp	Ref	Expression
1		intn3and.1	. . . 4 |- (ph -> -. ps)
2	1	intnanrd 694	. . 3 |- (ph -> -. (ps /\ ch))
3	2	intnanrd 694	. 2 |- (ph -> -. ((ps /\ ch) /\ th))
4		df-3an 777	. . 3 |- ((ps /\ ch /\ th) <-> ((ps /\ ch) /\ th))
5	4	negbii 187	. 2 |- (-. (ps /\ ch /\ th) <-> -. ((ps /\ ch) /\ th))
6	3, 5	sylibr 200	1 |- (ph -> -. (ps /\ ch /\ th))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   /\ w3a 775

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-an 225  df-3an 777

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