Theorem ifpbi23d 999

Description: Equivalence deduction for conditional operator for propositions. Convenience theorem for a frequent case. (Contributed by Wolf Lammen, 28-Apr-2024.)

Hypotheses

Ref	Expression
ifpbi23d.1	|- ( ph -> ( ch <-> et ) )
ifpbi23d.2	|- ( ph -> ( th <-> ze ) )

Assertion

Ref	Expression
ifpbi23d	|- ( ph -> (if- ( ps , ch , th ) <-> if- ( ps , et , ze ) ) )

Proof of Theorem ifpbi23d

Step	Hyp	Ref	Expression
1		biidd 172	. 2 |- ( ph -> ( ps <-> ps ) )
2		ifpbi23d.1	. 2 |- ( ph -> ( ch <-> et ) )
3		ifpbi23d.2	. 2 |- ( ph -> ( th <-> ze ) )
4	1, 2, 3	ifpbi123d 998	1 |- ( ph -> (if- ( ps , ch , th ) <-> if- ( ps , et , ze ) ) )

Syntax hints:   -> wi 4   <-> wb 105  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-in1 617  ax-in2 618  ax-io 714

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

This theorem is referenced by: (None)