Theorem if0ab 3622

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. (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		dfif6 3621	. 2 ⊢ if(𝜑, 𝐴, ∅) = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ ∅ ∣ ¬ 𝜑})
2		rab0 3536	. . 3 ⊢ {𝑥 ∈ ∅ ∣ ¬ 𝜑} = ∅
3	2	uneq2i 3369	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ ∅ ∣ ¬ 𝜑}) = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ ∅)
4		un0 3541	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ ∅) = {𝑥 ∈ 𝐴 ∣ 𝜑}
5	1, 3, 4	3eqtri 2257	1 ⊢ if(𝜑, 𝐴, ∅) = {𝑥 ∈ 𝐴 ∣ 𝜑}

Syntax hints:  ¬ wn 3   = wceq 1398  {crab 2524   ∪ cun 3208  ∅c0 3507  ifcif 3619

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rab 2529  df-v 2814  df-dif 3212  df-un 3214  df-nul 3508  df-if 3620

This theorem is referenced by:  if0ss  3623