Theorem if0ab 13340

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 3551, ⊢ (𝐴 ∈ 𝑉 → if(𝜑, 𝐴, ∅) ∈ 𝒫 𝐴), from which fmelpw1o 13341 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 3398	. . . . . 6 ⊢ ¬ 𝑥 ∈ ∅
2	1	intnanr 916	. . . . 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 2273	. 2 ⊢ {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ ∅ ∧ ¬ 𝜑))} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
6		df-if 3506	. 2 ⊢ if(𝜑, 𝐴, ∅) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ ∅ ∧ ¬ 𝜑))}
7		df-rab 2444	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
8	5, 6, 7	3eqtr4i 2188	1 ⊢ if(𝜑, 𝐴, ∅) = {𝑥 ∈ 𝐴 ∣ 𝜑}

Syntax hints:  ¬ wn 3   ∧ wa 103   ∨ wo 698   = wceq 1335   ∈ wcel 2128  {cab 2143  {crab 2439  ∅c0 3394  ifcif 3505

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rab 2444  df-v 2714  df-dif 3104  df-nul 3395  df-if 3506

This theorem is referenced by: (None)