Theorem ifbothdadc 3608

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 3580	. . . . . 6 |- ( ph -> if ( ph , A , B ) = A )
3	2	eqcomd 2212	. . . . 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 3583	. . . . . 6 |- ( -. ph -> if ( ph , A , B ) = B )
10	9	eqcomd 2212	. . . . 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 838	. . 3 |- (DECID ph -> ( ph \/ -. ph ) )
17	15, 16	syl 14	. 2 |- ( et -> ( ph \/ -. ph ) )
18	7, 14, 17	mpjaodan 800	1 |- ( et -> th )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 710  DECID wdc 836   = wceq 1373   ifcif 3575

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-dc 837  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-if 3576

This theorem is referenced by:  bitsinv1lem  12342  bitsinv1  12343  hashgcdeq  12632