Theorem iftrueb01 7572

Description: Using an if expression to represent a truth value by ∅ or 1_o. Unlike some theorems using if, 𝜑 does not need to be decidable. (Contributed by Jim Kingdon, 9-Jan-2026.)

Assertion

Ref	Expression
iftrueb01	⊢ (if(𝜑, 1o, ∅) = 1o ↔ 𝜑)

Proof of Theorem iftrueb01

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0lt1o 6703	. . . 4 ⊢ ∅ ∈ 1o
2		elex2 2838	. . . 4 ⊢ (∅ ∈ 1o → ∃𝑥 𝑥 ∈ 1o)
3	1, 2	ax-mp 5	. . 3 ⊢ ∃𝑥 𝑥 ∈ 1o
4		eleq2 2302	. . . . . 6 ⊢ (if(𝜑, 1o, ∅) = 1o → (𝑥 ∈ if(𝜑, 1o, ∅) ↔ 𝑥 ∈ 1o))
5		elif 3649	. . . . . . 7 ⊢ (𝑥 ∈ if(𝜑, 1o, ∅) ↔ ((𝜑 ∧ 𝑥 ∈ 1o) ∨ (¬ 𝜑 ∧ 𝑥 ∈ ∅)))
6		noel 3525	. . . . . . . . 9 ⊢ ¬ 𝑥 ∈ ∅
7	6	intnan 941	. . . . . . . 8 ⊢ ¬ (¬ 𝜑 ∧ 𝑥 ∈ ∅)
8	7	biorfi 758	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 1o) ↔ ((𝜑 ∧ 𝑥 ∈ 1o) ∨ (¬ 𝜑 ∧ 𝑥 ∈ ∅)))
9	5, 8	bitr4i 187	. . . . . 6 ⊢ (𝑥 ∈ if(𝜑, 1o, ∅) ↔ (𝜑 ∧ 𝑥 ∈ 1o))
10	4, 9	bitr3di 195	. . . . 5 ⊢ (if(𝜑, 1o, ∅) = 1o → (𝑥 ∈ 1o ↔ (𝜑 ∧ 𝑥 ∈ 1o)))
11		pm4.71r 394	. . . . 5 ⊢ ((𝑥 ∈ 1o → 𝜑) ↔ (𝑥 ∈ 1o ↔ (𝜑 ∧ 𝑥 ∈ 1o)))
12	10, 11	sylibr 134	. . . 4 ⊢ (if(𝜑, 1o, ∅) = 1o → (𝑥 ∈ 1o → 𝜑))
13	12	exlimdv 1872	. . 3 ⊢ (if(𝜑, 1o, ∅) = 1o → (∃𝑥 𝑥 ∈ 1o → 𝜑))
14	3, 13	mpi 15	. 2 ⊢ (if(𝜑, 1o, ∅) = 1o → 𝜑)
15		iftrue 3642	. 2 ⊢ (𝜑 → if(𝜑, 1o, ∅) = 1o)
16	14, 15	impbii 126	1 ⊢ (if(𝜑, 1o, ∅) = 1o ↔ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 720   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ∅c0 3520  ifcif 3635  1_oc1o 6670

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-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-nul 4254

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-dif 3222  df-un 3224  df-nul 3521  df-if 3636  df-sn 3711  df-suc 4511  df-1o 6677

This theorem is referenced by:  pw1map  16939