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Theorem if0ab 14179
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 3582, (𝐴𝑉 → if(𝜑, 𝐴, ∅) ∈ 𝒫 𝐴), from which fmelpw1o 14180 could be derived, yielding an alternative proof. (Contributed by BJ, 16-Aug-2024.)

Assertion
Ref Expression
if0ab if(𝜑, 𝐴, ∅) = {𝑥𝐴𝜑}
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥

Proof of Theorem if0ab
StepHypRef Expression
1 noel 3426 . . . . . 6 ¬ 𝑥 ∈ ∅
21intnanr 930 . . . . 5 ¬ (𝑥 ∈ ∅ ∧ ¬ 𝜑)
32biorfi 746 . . . 4 ((𝑥𝐴𝜑) ↔ ((𝑥𝐴𝜑) ∨ (𝑥 ∈ ∅ ∧ ¬ 𝜑)))
43bicomi 132 . . 3 (((𝑥𝐴𝜑) ∨ (𝑥 ∈ ∅ ∧ ¬ 𝜑)) ↔ (𝑥𝐴𝜑))
54abbii 2293 . 2 {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥 ∈ ∅ ∧ ¬ 𝜑))} = {𝑥 ∣ (𝑥𝐴𝜑)}
6 df-if 3535 . 2 if(𝜑, 𝐴, ∅) = {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥 ∈ ∅ ∧ ¬ 𝜑))}
7 df-rab 2464 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
85, 6, 73eqtr4i 2208 1 if(𝜑, 𝐴, ∅) = {𝑥𝐴𝜑}
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 104  wo 708   = wceq 1353  wcel 2148  {cab 2163  {crab 2459  c0 3422  ifcif 3534
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464  df-v 2739  df-dif 3131  df-nul 3423  df-if 3535
This theorem is referenced by: (None)
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