Theorem ifbothdc 3499

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 3474	. . . . . 6 |- ( ph -> if ( ph , A , B ) = A )
2	1	eqcomd 2143	. . . . 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 158	. . 3 |- ( ps -> ( ph -> th ) )
6	5	3ad2ant1 1002	. 2 |- ( ( ps /\ ch /\ DECID ph ) -> ( ph -> th ) )
7		iffalse 3477	. . . . . 6 |- ( -. ph -> if ( ph , A , B ) = B )
8	7	eqcomd 2143	. . . . 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 158	. . 3 |- ( ch -> ( -. ph -> th ) )
12	11	3ad2ant2 1003	. 2 |- ( ( ps /\ ch /\ DECID ph ) -> ( -. ph -> th ) )
13		exmiddc 821	. . 3 |- (DECID ph -> ( ph \/ -. ph ) )
14	13	3ad2ant3 1004	. 2 |- ( ( ps /\ ch /\ DECID ph ) -> ( ph \/ -. ph ) )
15	6, 12, 14	mpjaod 707	1 |- ( ( ps /\ ch /\ DECID ph ) -> th )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 104   \/ wo 697  DECID wdc 819   /\ w3a 962   = wceq 1331   ifcif 3469

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3an 964  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-if 3470

This theorem is referenced by:  isumlessdc  11258