Theorem 1fpid3 1029

Description: The value of the conditional operator for propositions is its third argument if the first and second argument imply the third argument. (Contributed by AV, 4-Apr-2021.)

Hypothesis

Ref	Expression
1fpid3.1	|- ( ( ph /\ ps ) -> ch )

Assertion

Ref	Expression
1fpid3	|- (if- ( ph , ps , ch ) -> ch )

Proof of Theorem 1fpid3

Step	Hyp	Ref	Expression
1		df-ifp 1013	. 2 |- (if- ( ph , ps , ch ) <-> ( ( ph /\ ps ) \/ ( -. ph /\ ch ) ) )
2		1fpid3.1	. . 3 |- ( ( ph /\ ps ) -> ch )
3		simpr 477	. . 3 |- ( ( -. ph /\ ch ) -> ch )
4	2, 3	jaoi 394	. 2 |- ( ( ( ph /\ ps ) \/ ( -. ph /\ ch ) ) -> ch )
5	1, 4	sylbi 207	1 |- (if- ( ph , ps , ch ) -> ch )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384  if-wif 1012

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013

This theorem is referenced by:  ifpsnprss  26518