Theorem ifbothdadc 3567

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 3540	. . . . . 6 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
3	2	eqcomd 2183	. . . . 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 3543	. . . . . 6 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
10	9	eqcomd 2183	. . . . 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 836	. . 3 ⊢ (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑))
17	15, 16	syl 14	. 2 ⊢ (𝜂 → (𝜑 ∨ ¬ 𝜑))
18	7, 14, 17	mpjaodan 798	1 ⊢ (𝜂 → 𝜃)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 708  DECID wdc 834   = wceq 1353  ifcif 3535

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 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-dc 835  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-if 3536

This theorem is referenced by:  hashgcdeq  12239