Theorem hb3an 1012

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 1009	. . 3 |- ((ph /\ ps) -> A.x(ph /\ ps))
4		hb.3	. . 3 |- (ch -> A.xch)
5	3, 4	hban 1009	. 2 |- (((ph /\ ps) /\ ch) -> A.x((ph /\ ps) /\ ch))
6		df-3an 777	. 2 |- ((ph /\ ps /\ ch) <-> ((ph /\ ps) /\ ch))
7	6	albii 999	. 2 |- (A.x(ph /\ ps /\ ch) <-> A.x((ph /\ ps) /\ ch))
8	5, 6, 7	3imtr4 219	1 |- ((ph /\ ps /\ ch) -> A.x(ph /\ ps /\ ch))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775  A.wal 954

This theorem is referenced by:  mopick2 1436

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 963  ax-4 973  ax-5o 975  ax-6o 978

This theorem depends on definitions:  df-bi 147  df-an 225  df-3an 777

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