Theorem ifbothdadc 3655

Description: A formula 𝜃 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(𝜑, 𝐴, 𝐵) → (𝜓 ↔ 𝜃))
ifbothdc.2	⊢ (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒 ↔ 𝜃))
ifbothdadc.3	⊢ ((𝜂 ∧ 𝜑) → 𝜓)
ifbothdadc.4	⊢ ((𝜂 ∧ ¬ 𝜑) → 𝜒)
ifbothdadc.dc	⊢ (𝜂 → DECID 𝜑)

Assertion

Ref	Expression
ifbothdadc	⊢ (𝜂 → 𝜃)

Proof of Theorem ifbothdadc

Step	Hyp	Ref	Expression
1		ifbothdadc.3	. . 3 ⊢ ((𝜂 ∧ 𝜑) → 𝜓)
2		iftrue 3626	. . . . . 6 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
3	2	eqcomd 2238	. . . . 5 ⊢ (𝜑 → 𝐴 = if(𝜑, 𝐴, 𝐵))
4		ifbothdc.1	. . . . 5 ⊢ (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓 ↔ 𝜃))
5	3, 4	syl 14	. . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜃))
6	5	adantl 277	. . 3 ⊢ ((𝜂 ∧ 𝜑) → (𝜓 ↔ 𝜃))
7	1, 6	mpbid 147	. 2 ⊢ ((𝜂 ∧ 𝜑) → 𝜃)
8		ifbothdadc.4	. . 3 ⊢ ((𝜂 ∧ ¬ 𝜑) → 𝜒)
9		iffalse 3629	. . . . . 6 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
10	9	eqcomd 2238	. . . . 5 ⊢ (¬ 𝜑 → 𝐵 = if(𝜑, 𝐴, 𝐵))
11		ifbothdc.2	. . . . 5 ⊢ (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒 ↔ 𝜃))
12	10, 11	syl 14	. . . 4 ⊢ (¬ 𝜑 → (𝜒 ↔ 𝜃))
13	12	adantl 277	. . 3 ⊢ ((𝜂 ∧ ¬ 𝜑) → (𝜒 ↔ 𝜃))
14	8, 13	mpbid 147	. 2 ⊢ ((𝜂 ∧ ¬ 𝜑) → 𝜃)
15		ifbothdadc.dc	. . 3 ⊢ (𝜂 → DECID 𝜑)
16		exmiddc 844	. . 3 ⊢ (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑))
17	15, 16	syl 14	. 2 ⊢ (𝜂 → (𝜑 ∨ ¬ 𝜑))
18	7, 14, 17	mpjaodan 806	1 ⊢ (𝜂 → 𝜃)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 716  DECID wdc 842   = wceq 1398  ifcif 3619

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-dc 843  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-if 3620

This theorem is referenced by:  bitsinv1lem  12640  bitsinv1  12641  hashgcdeq  12930  eulerpathprum  16462