Theorem ifbothdc 3590

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 3562	. . . . . 6 |- ( ph -> if ( ph , A , B ) = A )
2	1	eqcomd 2199	. . . . 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 159	. . 3 |- ( ps -> ( ph -> th ) )
6	5	3ad2ant1 1020	. 2 |- ( ( ps /\ ch /\ DECID ph ) -> ( ph -> th ) )
7		iffalse 3565	. . . . . 6 |- ( -. ph -> if ( ph , A , B ) = B )
8	7	eqcomd 2199	. . . . 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 159	. . 3 |- ( ch -> ( -. ph -> th ) )
12	11	3ad2ant2 1021	. 2 |- ( ( ps /\ ch /\ DECID ph ) -> ( -. ph -> th ) )
13		exmiddc 837	. . 3 |- (DECID ph -> ( ph \/ -. ph ) )
14	13	3ad2ant3 1022	. 2 |- ( ( ps /\ ch /\ DECID ph ) -> ( ph \/ -. ph ) )
15	6, 12, 14	mpjaod 719	1 |- ( ( ps /\ ch /\ DECID ph ) -> th )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   \/ wo 709  DECID wdc 835   /\ w3a 980   = wceq 1364   ifcif 3557

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 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-if 3558

This theorem is referenced by:  isumlessdc  11639  pcmptdvds  12483