Theorem bj-df-ifc 31735

Description: The definition of "ifc" if "if-" enters the main part. This is in line with the definition of a class as the extension of a predicate in df-clab 2597. (Contributed by BJ, 20-Sep-2019.)

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		bj-dfifc2 31734	. 2 ⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝜑 ∧ 𝑥 ∈ 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 ∈ 𝐵))}
2		df-ifp 1007	. . . 4 ⊢ (if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵) ↔ ((𝜑 ∧ 𝑥 ∈ 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 ∈ 𝐵)))
3	2	bicomi 213	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 ∈ 𝐵)) ↔ if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵))
4	3	abbii 2726	. 2 ⊢ {𝑥 ∣ ((𝜑 ∧ 𝑥 ∈ 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 ∈ 𝐵))} = {𝑥 ∣ if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵)}
5	1, 4	eqtri 2632	1 ⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵)}

Syntax hints:  ¬ wn 3   ∨ wo 382   ∧ wa 383  if-wif 1006   = wceq 1475   ∈ wcel 1977  {cab 2596  ifcif 4036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ifp 1007  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-if 4037

This theorem is referenced by:  bj-ififc  31736