Theorem dfif6

Description: An alternate definition of the conditional operator df-if as a simple class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013)

	Ref	Expression
Assertion	dfif6	⊢ if ( 𝜑 , 𝐴 , 𝐵 ) = ( { 𝑥 ∈ 𝐴 ∣ 𝜑 } ∪ { 𝑥 ∈ 𝐵 ∣ ¬ 𝜑 } )

Proof

Step	Hyp	Ref	Expression
1		eleq1w	⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴 ) )
2	1	anbi1d	⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ( 𝑦 ∈ 𝐴 ∧ 𝜑 ) ) )
3		eleq1w	⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵 ) )
4	3	anbi1d	⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 ∈ 𝐵 ∧ ¬ 𝜑 ) ↔ ( 𝑦 ∈ 𝐵 ∧ ¬ 𝜑 ) ) )
5	2 4	unabw	⊢ ( { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) } ∪ { 𝑥 ∣ ( 𝑥 ∈ 𝐵 ∧ ¬ 𝜑 ) } ) = { 𝑦 ∣ ( ( 𝑦 ∈ 𝐴 ∧ 𝜑 ) ∨ ( 𝑦 ∈ 𝐵 ∧ ¬ 𝜑 ) ) }
6		df-rab	⊢ { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) }
7		df-rab	⊢ { 𝑥 ∈ 𝐵 ∣ ¬ 𝜑 } = { 𝑥 ∣ ( 𝑥 ∈ 𝐵 ∧ ¬ 𝜑 ) }
8	6 7	uneq12i	⊢ ( { 𝑥 ∈ 𝐴 ∣ 𝜑 } ∪ { 𝑥 ∈ 𝐵 ∣ ¬ 𝜑 } ) = ( { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) } ∪ { 𝑥 ∣ ( 𝑥 ∈ 𝐵 ∧ ¬ 𝜑 ) } )
9		df-if	⊢ if ( 𝜑 , 𝐴 , 𝐵 ) = { 𝑦 ∣ ( ( 𝑦 ∈ 𝐴 ∧ 𝜑 ) ∨ ( 𝑦 ∈ 𝐵 ∧ ¬ 𝜑 ) ) }
10	5 8 9	3eqtr4ri	⊢ if ( 𝜑 , 𝐴 , 𝐵 ) = ( { 𝑥 ∈ 𝐴 ∣ 𝜑 } ∪ { 𝑥 ∈ 𝐵 ∣ ¬ 𝜑 } )