Theorem ifbothdc 3380

Description: A wff th containing a conditional operator is true when both of its cases are true. (Contributed by Jim Kingdon, 8-Aug-2021.)

Hypotheses

Ref	Expression
ifbothdc.1	|- ( A = if ( ph , A , B ) -> ( ps <-> th ) )
ifbothdc.2	|- ( B = if ( ph , A , B ) -> ( ch <-> th ) )

Assertion

Ref	Expression
ifbothdc	|- ( ( ps /\ ch /\ DECID ph ) -> th )

Proof of Theorem ifbothdc

Step	Hyp	Ref	Expression
1		iftrue 3356	. . . . . 6 |- ( ph -> if ( ph , A , B ) = A )
2	1	eqcomd 2086	. . . . 5 |- ( ph -> A = if ( ph , A , B ) )
3		ifbothdc.1	. . . . 5 |- ( A = if ( ph , A , B ) -> ( ps <-> th ) )
4	2, 3	syl 14	. . . 4 |- ( ph -> ( ps <-> th ) )
5	4	biimpcd 157	. . 3 |- ( ps -> ( ph -> th ) )
6	5	3ad2ant1 959	. 2 |- ( ( ps /\ ch /\ DECID ph ) -> ( ph -> th ) )
7		iffalse 3359	. . . . . 6 |- ( -. ph -> if ( ph , A , B ) = B )
8	7	eqcomd 2086	. . . . 5 |- ( -. ph -> B = if ( ph , A , B ) )
9		ifbothdc.2	. . . . 5 |- ( B = if ( ph , A , B ) -> ( ch <-> th ) )
10	8, 9	syl 14	. . . 4 |- ( -. ph -> ( ch <-> th ) )
11	10	biimpcd 157	. . 3 |- ( ch -> ( -. ph -> th ) )
12	11	3ad2ant2 960	. 2 |- ( ( ps /\ ch /\ DECID ph ) -> ( -. ph -> th ) )
13		exmiddc 777	. . 3 |- (DECID ph -> ( ph \/ -. ph ) )
14	13	3ad2ant3 961	. 2 |- ( ( ps /\ ch /\ DECID ph ) -> ( ph \/ -. ph ) )
15	6, 12, 14	mpjaod 670	1 |- ( ( ps /\ ch /\ DECID ph ) -> th )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 103   \/ wo 661  DECID wdc 775   /\ w3a 919   = wceq 1284   ifcif 3351

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3an 921  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-if 3352

This theorem is referenced by: (None)