Theorem 1fpid3 1000

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 984	. 2 |- (if- ( ph , ps , ch ) <-> ( ( ph /\ ps ) \/ ( -. ph /\ ch ) ) )
2		1fpid3.1	. . 3 |- ( ( ph /\ ps ) -> ch )
3		simpr 110	. . 3 |- ( ( -. ph /\ ch ) -> ch )
4	2, 3	jaoi 721	. 2 |- ( ( ( ph /\ ps ) \/ ( -. ph /\ ch ) ) -> ch )
5	1, 4	sylbi 121	1 |- (if- ( ph , ps , ch ) -> ch )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 713  if-wif 983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714

This theorem depends on definitions:  df-bi 117  df-ifp 984

This theorem is referenced by: (None)