Theorem hb3or 706

Description: If x is not free in ph, ps, and ch, it is not free in (ph \/ ps \/ ch).

Hypotheses

Ref	Expression
hb.1	|- (ph -> A.xph)
hb.2	|- (ps -> A.xps)
hb.3	|- (ch -> A.xch)

Assertion

Ref	Expression
hb3or	|- ((ph \/ ps \/ ch) -> A.x(ph \/ ps \/ ch))

Proof of Theorem hb3or

Step	Hyp	Ref	Expression
1		hb.1	. . . 4 |- (ph -> A.xph)
2		hb.2	. . . 4 |- (ps -> A.xps)
3	1, 2	hbor 703	. . 3 |- ((ph \/ ps) -> A.x(ph \/ ps))
4		hb.3	. . 3 |- (ch -> A.xch)
5	3, 4	hbor 703	. 2 |- (((ph \/ ps) \/ ch) -> A.x((ph \/ ps) \/ ch))
6		df-3or 582	. 2 |- ((ph \/ ps \/ ch) <-> ((ph \/ ps) \/ ch))
7	6	bial 695	. 2 |- (A.x(ph \/ ps \/ ch) <-> A.x((ph \/ ps) \/ ch))
8	5, 6, 7	3imtr4 192	1 |- ((ph \/ ps \/ ch) -> A.x(ph \/ ps \/ ch))

Syntax hints:   -> wi 2   \/ wo 195   \/ w3o 580  A.wal 672

This theorem is referenced by:  hbrdg 2974

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582

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