Theorem ifbothdc 3357

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 3336	. . . . . 6 |- ( ph -> if ( ph , A , B ) = A )
2	1	eqcomd 2045	. . . . 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 148	. . 3 |- ( ps -> ( ph -> th ) )
6	5	3ad2ant1 925	. 2 |- ( ( ps /\ ch /\ DECID ph ) -> ( ph -> th ) )
7		iffalse 3339	. . . . . 6 |- ( -. ph -> if ( ph , A , B ) = B )
8	7	eqcomd 2045	. . . . 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 148	. . 3 |- ( ch -> ( -. ph -> th ) )
12	11	3ad2ant2 926	. 2 |- ( ( ps /\ ch /\ DECID ph ) -> ( -. ph -> th ) )
13		exmiddc 744	. . 3 |- (DECID ph -> ( ph \/ -. ph ) )
14	13	3ad2ant3 927	. 2 |- ( ( ps /\ ch /\ DECID ph ) -> ( ph \/ -. ph ) )
15	6, 12, 14	mpjaod 638	1 |- ( ( ps /\ ch /\ DECID ph ) -> th )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 98   \/ wo 629  DECID wdc 742   /\ w3a 885   = wceq 1243   ifcif 3331

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3an 887  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-if 3332

This theorem is referenced by: (None)