Theorem bj-ififc 32566

Description: A theorem linking if- and if. (Contributed by BJ, 24-Sep-2019.)

Assertion

Ref	Expression
bj-ififc	⊢ (𝑥 ∈ if(𝜑, 𝐴, 𝐵) ↔ if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵))

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

Proof of Theorem bj-ififc

Step	Hyp	Ref	Expression
1		bj-df-ifc 32565	. 2 ⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵)}
2	1	abeq2i 2735	1 ⊢ (𝑥 ∈ if(𝜑, 𝐴, 𝐵) ↔ if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵))

Syntax hints:   ↔ wb 196  if-wif 1012   ∈ wcel 1990  ifcif 4086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-if 4087

This theorem is referenced by: (None)