Theorem bj-df-ifc 36562

Description: Candidate definition for the conditional operator for classes. This is in line with the definition of a class as the extension of a predicate in df-clab 2712. We reprove the current df-if 4531 from it in bj-dfif 36563. (Contributed by BJ, 20-Sep-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-df-ifc	⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵)}

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

Proof of Theorem bj-df-ifc

Step	Hyp	Ref	Expression
1		df-if 4531	. 2 ⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))}
2		ancom 460	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝜑 ∧ 𝑥 ∈ 𝐴))
3		ancom 460	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ ¬ 𝜑) ↔ (¬ 𝜑 ∧ 𝑥 ∈ 𝐵))
4	2, 3	orbi12i 914	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑)) ↔ ((𝜑 ∧ 𝑥 ∈ 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 ∈ 𝐵)))
5		df-ifp 1063	. . . 4 ⊢ (if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵) ↔ ((𝜑 ∧ 𝑥 ∈ 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 ∈ 𝐵)))
6	4, 5	bitr4i 278	. . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑)) ↔ if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵))
7	6	abbii 2806	. 2 ⊢ {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))} = {𝑥 ∣ if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵)}
8	1, 7	eqtri 2762	1 ⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵)}

Syntax hints:  ¬ wn 3   ∧ wa 395   ∨ wo 847  if-wif 1062   = wceq 1536   ∈ wcel 2105  {cab 2711  ifcif 4530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-9 2115  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-if 4531

This theorem is referenced by:  bj-dfif  36563  bj-ififc  36564