Theorem if0ab 16422

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 3660, ⊢ (𝐴 ∈ 𝑉 → if(𝜑, 𝐴, ∅) ∈ 𝒫 𝐴), from which fmelpw1o 7465 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 3498	. . . . . 6 ⊢ ¬ 𝑥 ∈ ∅
2	1	intnanr 937	. . . . 5 ⊢ ¬ (𝑥 ∈ ∅ ∧ ¬ 𝜑)
3	2	biorfi 753	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ ∅ ∧ ¬ 𝜑)))
4	3	bicomi 132	. . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ ∅ ∧ ¬ 𝜑)) ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))
5	4	abbii 2347	. 2 ⊢ {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ ∅ ∧ ¬ 𝜑))} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
6		df-if 3606	. 2 ⊢ if(𝜑, 𝐴, ∅) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ ∅ ∧ ¬ 𝜑))}
7		df-rab 2519	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
8	5, 6, 7	3eqtr4i 2262	1 ⊢ if(𝜑, 𝐴, ∅) = {𝑥 ∈ 𝐴 ∣ 𝜑}

Syntax hints:  ¬ wn 3   ∧ wa 104   ∨ wo 715   = wceq 1397   ∈ wcel 2202  {cab 2217  {crab 2514  ∅c0 3494  ifcif 3605

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rab 2519  df-v 2804  df-dif 3202  df-nul 3495  df-if 3606

This theorem is referenced by: (None)