Theorem hb3an 707

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
hb3an	|- ((ph /\ ps /\ ch) -> A.x(ph /\ ps /\ ch))

Proof of Theorem hb3an

Step	Hyp	Ref	Expression
1		hb.1	. . . 4 |- (ph -> A.xph)
2		hb.2	. . . 4 |- (ps -> A.xps)
3	1, 2	hban 704	. . 3 |- ((ph /\ ps) -> A.x(ph /\ ps))
4		hb.3	. . 3 |- (ch -> A.xch)
5	3, 4	hban 704	. 2 |- (((ph /\ ps) /\ ch) -> A.x((ph /\ ps) /\ ch))
6		df-3an 583	. 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   /\ wa 196   /\ w3a 581  A.wal 672

This theorem is referenced by:  mopick2 1057

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-an 198  df-3an 583

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