Theorem ifp2 986

Description: Forward direction of dfifp2dc 987. This direction does not require decidability. (Contributed by Jim Kingdon, 25-Jan-2026.)

Assertion

Ref	Expression
ifp2	|- (if- ( ph , ps , ch ) -> ( ( ph -> ps ) /\ ( -. ph -> ch ) ) )

Proof of Theorem ifp2

Step	Hyp	Ref	Expression
1		df-ifp 984	. 2 |- (if- ( ph , ps , ch ) <-> ( ( ph /\ ps ) \/ ( -. ph /\ ch ) ) )
2		pm3.4 333	. . . 4 |- ( ( ph /\ ps ) -> ( ph -> ps ) )
3		pm2.24 624	. . . . 5 |- ( ph -> ( -. ph -> ch ) )
4	3	adantr 276	. . . 4 |- ( ( ph /\ ps ) -> ( -. ph -> ch ) )
5	2, 4	jca 306	. . 3 |- ( ( ph /\ ps ) -> ( ( ph -> ps ) /\ ( -. ph -> ch ) ) )
6		ax-in2 618	. . . . 5 |- ( -. ph -> ( ph -> ps ) )
7	6	adantr 276	. . . 4 |- ( ( -. ph /\ ch ) -> ( ph -> ps ) )
8		pm3.4 333	. . . 4 |- ( ( -. ph /\ ch ) -> ( -. ph -> ch ) )
9	7, 8	jca 306	. . 3 |- ( ( -. ph /\ ch ) -> ( ( ph -> ps ) /\ ( -. ph -> ch ) ) )
10	5, 9	jaoi 721	. 2 |- ( ( ( ph /\ ps ) \/ ( -. ph /\ ch ) ) -> ( ( ph -> ps ) /\ ( -. ph -> ch ) ) )
11	1, 10	sylbi 121	1 |- (if- ( ph , ps , ch ) -> ( ( ph -> ps ) /\ ( -. ph -> 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-in2 618  ax-io 714

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

This theorem is referenced by:  dfifp2dc  987