Theorem hbor 703

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

Hypotheses

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

Assertion

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

Proof of Theorem hbor

Step	Hyp	Ref	Expression
1		hb.1	. . . 4 |- (ph -> A.xph)
2	1	hbne 699	. . 3 |- (-. ph -> A.x -. ph)
3		hb.2	. . 3 |- (ps -> A.xps)
4	2, 3	hbim 702	. 2 |- ((-. ph -> ps) -> A.x(-. ph -> ps))
5		df-or 197	. 2 |- ((ph \/ ps) <-> (-. ph -> ps))
6	5	bial 695	. 2 |- (A.x(ph \/ ps) <-> A.x(-. ph -> ps))
7	4, 5, 6	3imtr4 192	1 |- ((ph \/ ps) -> A.x(ph \/ ps))

Syntax hints:  -. wn 1   -> wi 2   \/ wo 195  A.wal 672

This theorem is referenced by:  hb3or 706  hbun 1614  hbpr 1824  hbsuc 2294

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

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