Theorem ifpbi123

Description: Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020)

	Ref	Expression
Assertion	ifpbi123	|- ( ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) -> ( if- ( ph , ch , ta ) <-> if- ( ps , th , et ) ) )

Proof

Step	Hyp	Ref	Expression
1		simp1	|- ( ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) -> ( ph <-> ps ) )
2		simp2	|- ( ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) -> ( ch <-> th ) )
3		simp3	|- ( ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) -> ( ta <-> et ) )
4	1 2 3	ifpbi123d	|- ( ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) -> ( if- ( ph , ch , ta ) <-> if- ( ps , th , et ) ) )