Theorem dfif3

Description: Alternate definition of the conditional operator df-if . Note that ph is independent of x i.e. a constant true or false. (Contributed by NM, 25-Aug-2013) (Revised by Mario Carneiro, 8-Sep-2013)

	Ref	Expression
Hypothesis	dfif3.1	⊢ 𝐶 = { 𝑥 ∣ 𝜑 }
Assertion	dfif3	⊢ if ( 𝜑 , 𝐴 , 𝐵 ) = ( ( 𝐴 ∩ 𝐶 ) ∪ ( 𝐵 ∩ ( V ∖ 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		dfif3.1	⊢ 𝐶 = { 𝑥 ∣ 𝜑 }
2		dfif6	⊢ if ( 𝜑 , 𝐴 , 𝐵 ) = ( { 𝑦 ∈ 𝐴 ∣ 𝜑 } ∪ { 𝑦 ∈ 𝐵 ∣ ¬ 𝜑 } )
3		biidd	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜑 ) )
4	3	cbvabv	⊢ { 𝑥 ∣ 𝜑 } = { 𝑦 ∣ 𝜑 }
5	1 4	eqtri	⊢ 𝐶 = { 𝑦 ∣ 𝜑 }
6	5	ineq2i	⊢ ( 𝐴 ∩ 𝐶 ) = ( 𝐴 ∩ { 𝑦 ∣ 𝜑 } )
7		dfrab3	⊢ { 𝑦 ∈ 𝐴 ∣ 𝜑 } = ( 𝐴 ∩ { 𝑦 ∣ 𝜑 } )
8	6 7	eqtr4i	⊢ ( 𝐴 ∩ 𝐶 ) = { 𝑦 ∈ 𝐴 ∣ 𝜑 }
9		dfrab3	⊢ { 𝑦 ∈ 𝐵 ∣ ¬ 𝜑 } = ( 𝐵 ∩ { 𝑦 ∣ ¬ 𝜑 } )
10		biidd	⊢ ( 𝑦 = 𝑧 → ( 𝜑 ↔ 𝜑 ) )
11	10	notabw	⊢ { 𝑦 ∣ ¬ 𝜑 } = ( V ∖ { 𝑧 ∣ 𝜑 } )
12		biidd	⊢ ( 𝑥 = 𝑧 → ( 𝜑 ↔ 𝜑 ) )
13	12	cbvabv	⊢ { 𝑥 ∣ 𝜑 } = { 𝑧 ∣ 𝜑 }
14	1 13	eqtri	⊢ 𝐶 = { 𝑧 ∣ 𝜑 }
15	14	difeq2i	⊢ ( V ∖ 𝐶 ) = ( V ∖ { 𝑧 ∣ 𝜑 } )
16	11 15	eqtr4i	⊢ { 𝑦 ∣ ¬ 𝜑 } = ( V ∖ 𝐶 )
17	16	ineq2i	⊢ ( 𝐵 ∩ { 𝑦 ∣ ¬ 𝜑 } ) = ( 𝐵 ∩ ( V ∖ 𝐶 ) )
18	9 17	eqtr2i	⊢ ( 𝐵 ∩ ( V ∖ 𝐶 ) ) = { 𝑦 ∈ 𝐵 ∣ ¬ 𝜑 }
19	8 18	uneq12i	⊢ ( ( 𝐴 ∩ 𝐶 ) ∪ ( 𝐵 ∩ ( V ∖ 𝐶 ) ) ) = ( { 𝑦 ∈ 𝐴 ∣ 𝜑 } ∪ { 𝑦 ∈ 𝐵 ∣ ¬ 𝜑 } )
20	2 19	eqtr4i	⊢ if ( 𝜑 , 𝐴 , 𝐵 ) = ( ( 𝐴 ∩ 𝐶 ) ∪ ( 𝐵 ∩ ( V ∖ 𝐶 ) ) )