Theorem if0ab 3607

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 3606	. 2 ⊢ if(𝜑, 𝐴, ∅) = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ ∅ ∣ ¬ 𝜑})
2		rab0 3522	. . 3 ⊢ {𝑥 ∈ ∅ ∣ ¬ 𝜑} = ∅
3	2	uneq2i 3357	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ ∅ ∣ ¬ 𝜑}) = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ ∅)
4		un0 3527	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ ∅) = {𝑥 ∈ 𝐴 ∣ 𝜑}
5	1, 3, 4	3eqtri 2255	1 ⊢ if(𝜑, 𝐴, ∅) = {𝑥 ∈ 𝐴 ∣ 𝜑}

Syntax hints:  ¬ wn 3   = wceq 1397  {crab 2513   ∪ cun 3197  ∅c0 3493  ifcif 3604

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 2212

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-rab 2518  df-v 2803  df-dif 3201  df-un 3203  df-nul 3494  df-if 3605

This theorem is referenced by:  if0ss  3608