Theorem hbimtg 13873

Description: A more general and closed form of hbim 1354.

Assertion

Ref	Expression
hbimtg	|- ((A.x(ph -> A.xch) /\ (ps -> A.xth)) -> ((ch -> ps) -> A.x(ph -> th)))

Proof of Theorem hbimtg

Step	Hyp	Ref	Expression
1		hbntg 13872	. . . 4 |- (A.x(ph -> A.xch) -> (-. ch -> A.x -. ph))
2		pm2.21 92	. . . . 5 |- (-. ph -> (ph -> th))
3	2	alimi 1338	. . . 4 |- (A.x -. ph -> A.x(ph -> th))
4	1, 3	syl6 25	. . 3 |- (A.x(ph -> A.xch) -> (-. ch -> A.x(ph -> th)))
5	4	adantr 425	. 2 |- ((A.x(ph -> A.xch) /\ (ps -> A.xth)) -> (-. ch -> A.x(ph -> th)))
6		ax-1 4	. . . . 5 |- (th -> (ph -> th))
7	6	alimi 1338	. . . 4 |- (A.xth -> A.x(ph -> th))
8	7	imim2i 11	. . 3 |- ((ps -> A.xth) -> (ps -> A.x(ph -> th)))
9	8	adantl 424	. 2 |- ((A.x(ph -> A.xch) /\ (ps -> A.xth)) -> (ps -> A.x(ph -> th)))
10	5, 9	jad 156	1 |- ((A.x(ph -> A.xch) /\ (ps -> A.xth)) -> ((ch -> ps) -> A.x(ph -> th)))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240  A.wal 1296

This theorem is referenced by:  hbimg 13876

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1305  ax-4 1319  ax-5o 1321  ax-6o 1324

This theorem depends on definitions:  df-bi 164  df-an 242

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