Theorem ifbothdadc 3639

Description: A formula th containing a decidable conditional operator is true when both of its cases are true. (Contributed by Jim Kingdon, 3-Jun-2022.)

Hypotheses

Ref	Expression
ifbothdc.1	|- ( A = if ( ph , A , B ) -> ( ps <-> th ) )
ifbothdc.2	|- ( B = if ( ph , A , B ) -> ( ch <-> th ) )
ifbothdadc.3	|- ( ( et /\ ph ) -> ps )
ifbothdadc.4	|- ( ( et /\ -. ph ) -> ch )
ifbothdadc.dc	|- ( et -> DECID ph )

Assertion

Ref	Expression
ifbothdadc	|- ( et -> th )

Proof of Theorem ifbothdadc

Step	Hyp	Ref	Expression
1		ifbothdadc.3	. . 3 |- ( ( et /\ ph ) -> ps )
2		iftrue 3610	. . . . . 6 |- ( ph -> if ( ph , A , B ) = A )
3	2	eqcomd 2237	. . . . 5 |- ( ph -> A = if ( ph , A , B ) )
4		ifbothdc.1	. . . . 5 |- ( A = if ( ph , A , B ) -> ( ps <-> th ) )
5	3, 4	syl 14	. . . 4 |- ( ph -> ( ps <-> th ) )
6	5	adantl 277	. . 3 |- ( ( et /\ ph ) -> ( ps <-> th ) )
7	1, 6	mpbid 147	. 2 |- ( ( et /\ ph ) -> th )
8		ifbothdadc.4	. . 3 |- ( ( et /\ -. ph ) -> ch )
9		iffalse 3613	. . . . . 6 |- ( -. ph -> if ( ph , A , B ) = B )
10	9	eqcomd 2237	. . . . 5 |- ( -. ph -> B = if ( ph , A , B ) )
11		ifbothdc.2	. . . . 5 |- ( B = if ( ph , A , B ) -> ( ch <-> th ) )
12	10, 11	syl 14	. . . 4 |- ( -. ph -> ( ch <-> th ) )
13	12	adantl 277	. . 3 |- ( ( et /\ -. ph ) -> ( ch <-> th ) )
14	8, 13	mpbid 147	. 2 |- ( ( et /\ -. ph ) -> th )
15		ifbothdadc.dc	. . 3 |- ( et -> DECID ph )
16		exmiddc 843	. . 3 |- (DECID ph -> ( ph \/ -. ph ) )
17	15, 16	syl 14	. 2 |- ( et -> ( ph \/ -. ph ) )
18	7, 14, 17	mpjaodan 805	1 |- ( et -> th )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 715  DECID wdc 841   = wceq 1397   ifcif 3605

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 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-dc 842  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-if 3606

This theorem is referenced by:  bitsinv1lem  12527  bitsinv1  12528  hashgcdeq  12817  eulerpathprum  16337