Theorem if0ab 13687

Description: Expression of a conditional class as a class abstraction when the False alternative is the empty class: in that case, the conditional class is the extension, in the True alternative, of the condition.

Remark: a consequence which could be formalized is the inclusion ⊢ if(𝜑, 𝐴, ∅) ⊆ 𝐴 and therefore, using elpwg 3567, ⊢ (𝐴 ∈ 𝑉 → if(𝜑, 𝐴, ∅) ∈ 𝒫 𝐴), from which fmelpw1o 13688 could be derived, yielding an alternative proof. (Contributed by BJ, 16-Aug-2024.)

Assertion

Ref	Expression
if0ab	⊢ if(𝜑, 𝐴, ∅) = {𝑥 ∈ 𝐴 ∣ 𝜑}

Distinct variable groups:   𝑥,𝐴   𝜑,𝑥

Proof of Theorem if0ab

Step	Hyp	Ref	Expression
1		noel 3413	. . . . . 6 ⊢ ¬ 𝑥 ∈ ∅
2	1	intnanr 920	. . . . 5 ⊢ ¬ (𝑥 ∈ ∅ ∧ ¬ 𝜑)
3	2	biorfi 736	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ ∅ ∧ ¬ 𝜑)))
4	3	bicomi 131	. . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ ∅ ∧ ¬ 𝜑)) ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))
5	4	abbii 2282	. 2 ⊢ {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ ∅ ∧ ¬ 𝜑))} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
6		df-if 3521	. 2 ⊢ if(𝜑, 𝐴, ∅) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ ∅ ∧ ¬ 𝜑))}
7		df-rab 2453	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
8	5, 6, 7	3eqtr4i 2196	1 ⊢ if(𝜑, 𝐴, ∅) = {𝑥 ∈ 𝐴 ∣ 𝜑}

Syntax hints:  ¬ wn 3   ∧ wa 103   ∨ wo 698   = wceq 1343   ∈ wcel 2136  {cab 2151  {crab 2448  ∅c0 3409  ifcif 3520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rab 2453  df-v 2728  df-dif 3118  df-nul 3410  df-if 3521

This theorem is referenced by: (None)