Theorem ifbothdadc 3605

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 3577	. . . . . 6 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
3	2	eqcomd 2212	. . . . 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 3580	. . . . . 6 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
10	9	eqcomd 2212	. . . . 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 838	. . 3 ⊢ (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑))
17	15, 16	syl 14	. 2 ⊢ (𝜂 → (𝜑 ∨ ¬ 𝜑))
18	7, 14, 17	mpjaodan 800	1 ⊢ (𝜂 → 𝜃)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 710  DECID wdc 836   = wceq 1373  ifcif 3572

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 3573

This theorem is referenced by:  bitsinv1lem  12316  bitsinv1  12317  hashgcdeq  12606