Theorem impbida 517

Description: Deduce an equivalence from two implications.

Hypotheses

Ref	Expression
impbida.1	|- ((ph /\ ps) -> ch)
impbida.2	|- ((ph /\ ch) -> ps)

Assertion

Ref	Expression
impbida	|- (ph -> (ps <-> ch))

Proof of Theorem impbida

Step	Hyp	Ref	Expression
1		impbida.1	. . 3 |- ((ph /\ ps) -> ch)
2	1	ex 373	. 2 |- (ph -> (ps -> ch))
3		impbida.2	. . 3 |- ((ph /\ ch) -> ps)
4	3	ex 373	. 2 |- (ph -> (ch -> ps))
5	2, 4	impbid 514	1 |- (ph -> (ps <-> ch))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223

This theorem is referenced by:  eqop 4088  en3d 4382  elcls 7646  iscncl 7709  metcnp 7826  cmsss 7931  grpinvid1 8006  grpinvid2 8007  leopmult 9979  hst1ht 10064  cnfilca 10451  imonclem 10583

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

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

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