Theorem bj-dfif 35924

Description: Alternate definition of the conditional operator for classes, which used to be the main definition. (Contributed by BJ, 26-Dec-2023.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-dfif	⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝜑 ∧ 𝑥 ∈ 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 ∈ 𝐵))}

Distinct variable groups:   𝜑,𝑥   𝑥,𝐴   𝑥,𝐵

Proof of Theorem bj-dfif

Step	Hyp	Ref	Expression
1		bj-df-ifc 35923	. 2 ⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵)}
2		df-ifp 1061	. . 3 ⊢ (if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵) ↔ ((𝜑 ∧ 𝑥 ∈ 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 ∈ 𝐵)))
3	2	abbii 2801	. 2 ⊢ {𝑥 ∣ if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵)} = {𝑥 ∣ ((𝜑 ∧ 𝑥 ∈ 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 ∈ 𝐵))}
4	1, 3	eqtri 2759	1 ⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝜑 ∧ 𝑥 ∈ 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 ∈ 𝐵))}

Syntax hints:  ¬ wn 3   ∧ wa 395   ∨ wo 844  if-wif 1060   = wceq 1540   ∈ wcel 2105  {cab 2708  ifcif 4528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ifp 1061  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-if 4529

This theorem is referenced by: (None)