Theorem bj-df-ifc 32690

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 2638. (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 32689	. 2 ⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝜑 ∧ 𝑥 ∈ 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 ∈ 𝐵))}
2		df-ifp 1033	. . . 4 ⊢ (if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵) ↔ ((𝜑 ∧ 𝑥 ∈ 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 ∈ 𝐵)))
3	2	bicomi 214	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 ∈ 𝐵)) ↔ if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵))
4	3	abbii 2768	. 2 ⊢ {𝑥 ∣ ((𝜑 ∧ 𝑥 ∈ 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 ∈ 𝐵))} = {𝑥 ∣ if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵)}
5	1, 4	eqtri 2673	1 ⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵)}

Syntax hints:  ¬ wn 3   ∨ wo 382   ∧ wa 383  if-wif 1032   = wceq 1523   ∈ wcel 2030  {cab 2637  ifcif 4119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ifp 1033  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-if 4120

This theorem is referenced by:  bj-ififc  32691