Theorem if0ab 3641

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 3640	. 2 ⊢ if(𝜑, 𝐴, ∅) = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ ∅ ∣ ¬ 𝜑})
2		rab0 3551	. . 3 ⊢ {𝑥 ∈ ∅ ∣ ¬ 𝜑} = ∅
3	2	uneq2i 3380	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ ∅ ∣ ¬ 𝜑}) = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ ∅)
4		un0 3556	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ ∅) = {𝑥 ∈ 𝐴 ∣ 𝜑}
5	1, 3, 4	3eqtri 2263	1 ⊢ if(𝜑, 𝐴, ∅) = {𝑥 ∈ 𝐴 ∣ 𝜑}

Syntax hints:  ¬ wn 3   = wceq 1402  {crab 2532   ∪ cun 3218  ∅c0 3520  ifcif 3638

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-nul 3521  df-if 3639

This theorem is referenced by:  if0ss  3642