Theorem ifbothdc 3637

Description: A wff 𝜃 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(𝜑, 𝐴, 𝐵) → (𝜓 ↔ 𝜃))
ifbothdc.2	⊢ (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒 ↔ 𝜃))

Assertion

Ref	Expression
ifbothdc	⊢ ((𝜓 ∧ 𝜒 ∧ DECID 𝜑) → 𝜃)

Proof of Theorem ifbothdc

Step	Hyp	Ref	Expression
1		iftrue 3607	. . . . . 6 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
2	1	eqcomd 2235	. . . . 5 ⊢ (𝜑 → 𝐴 = if(𝜑, 𝐴, 𝐵))
3		ifbothdc.1	. . . . 5 ⊢ (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓 ↔ 𝜃))
4	2, 3	syl 14	. . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜃))
5	4	biimpcd 159	. . 3 ⊢ (𝜓 → (𝜑 → 𝜃))
6	5	3ad2ant1 1042	. 2 ⊢ ((𝜓 ∧ 𝜒 ∧ DECID 𝜑) → (𝜑 → 𝜃))
7		iffalse 3610	. . . . . 6 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
8	7	eqcomd 2235	. . . . 5 ⊢ (¬ 𝜑 → 𝐵 = if(𝜑, 𝐴, 𝐵))
9		ifbothdc.2	. . . . 5 ⊢ (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒 ↔ 𝜃))
10	8, 9	syl 14	. . . 4 ⊢ (¬ 𝜑 → (𝜒 ↔ 𝜃))
11	10	biimpcd 159	. . 3 ⊢ (𝜒 → (¬ 𝜑 → 𝜃))
12	11	3ad2ant2 1043	. 2 ⊢ ((𝜓 ∧ 𝜒 ∧ DECID 𝜑) → (¬ 𝜑 → 𝜃))
13		exmiddc 841	. . 3 ⊢ (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑))
14	13	3ad2ant3 1044	. 2 ⊢ ((𝜓 ∧ 𝜒 ∧ DECID 𝜑) → (𝜑 ∨ ¬ 𝜑))
15	6, 12, 14	mpjaod 723	1 ⊢ ((𝜓 ∧ 𝜒 ∧ DECID 𝜑) → 𝜃)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105   ∨ wo 713  DECID wdc 839   ∧ w3a 1002   = wceq 1395  ifcif 3602

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 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3an 1004  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-if 3603

This theorem is referenced by:  isumlessdc  12015  pcmptdvds  12876