Theorem hbbid 789

Description: Deduction form of bound-variable hypothesis builder hbbi 705.

Hypotheses

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

Assertion

Ref	Expression
hbbid	|- (ph -> ((ps <-> ch) -> A.x(ps <-> ch)))

Proof of Theorem hbbid

Step	Hyp	Ref	Expression
1		hbbid.1	. . . 4 |- (ph -> A.xph)
2		hbbid.2	. . . 4 |- (ph -> (ps -> A.xps))
3		hbbid.3	. . . 4 |- (ph -> (ch -> A.xch))
4	1, 2, 3	hbimd 787	. . 3 |- (ph -> ((ps -> ch) -> A.x(ps -> ch)))
5	1, 3, 2	hbimd 787	. . 3 |- (ph -> ((ch -> ps) -> A.x(ch -> ps)))
6	4, 5	anim12d 431	. 2 |- (ph -> (((ps -> ch) /\ (ch -> ps)) -> (A.x(ps -> ch) /\ A.x(ch -> ps))))
7		bi 396	. 2 |- ((ps <-> ch) <-> ((ps -> ch) /\ (ch -> ps)))
8		albi 785	. 2 |- (A.x(ps <-> ch) <-> (A.x(ps -> ch) /\ A.x(ch -> ps)))
9	6, 7, 8	3imtr4g 426	1 |- (ph -> ((ps <-> ch) -> A.x(ps <-> ch)))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672

This theorem is referenced by:  hbeu 1016  axextnd 3737  axrepndlem1 3738  axrepndlem2 3739  axacndlem4 3756  axacndlem5 3757  axacnd 3758

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

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