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	$$⊢C=\left\{x|\phi \right\}$$
Assertion	dfif3	$$⊢if\left(\phi ,A,B\right)=\left(A\cap C\right)\cup \left(B\cap \left(\mathrm{V}\setminus C\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		dfif3.1	$$⊢C=\left\{x|\phi \right\}$$
2		dfif6	$$⊢if\left(\phi ,A,B\right)=\left\{y\in A|\phi \right\}\cup \left\{y\in B|\neg \phi \right\}$$
3		biidd	$$⊢x=y\to \left(\phi \leftrightarrow \phi \right)$$
4	3	cbvabv	$$⊢\left\{x|\phi \right\}=\left\{y|\phi \right\}$$
5	1 4	eqtri	$$⊢C=\left\{y|\phi \right\}$$
6	5	ineq2i	$$⊢A\cap C=A\cap \left\{y|\phi \right\}$$
7		dfrab3	$$⊢\left\{y\in A|\phi \right\}=A\cap \left\{y|\phi \right\}$$
8	6 7	eqtr4i	$$⊢A\cap C=\left\{y\in A|\phi \right\}$$
9		dfrab3	$$⊢\left\{y\in B|\neg \phi \right\}=B\cap \left\{y|\neg \phi \right\}$$
10		biidd	$$⊢y=z\to \left(\phi \leftrightarrow \phi \right)$$
11	10	notabw	$$⊢\left\{y|\neg \phi \right\}=\mathrm{V}\setminus \left\{z|\phi \right\}$$
12		biidd	$$⊢x=z\to \left(\phi \leftrightarrow \phi \right)$$
13	12	cbvabv	$$⊢\left\{x|\phi \right\}=\left\{z|\phi \right\}$$
14	1 13	eqtri	$$⊢C=\left\{z|\phi \right\}$$
15	14	difeq2i	$$⊢\mathrm{V}\setminus C=\mathrm{V}\setminus \left\{z|\phi \right\}$$
16	11 15	eqtr4i	$$⊢\left\{y|\neg \phi \right\}=\mathrm{V}\setminus C$$
17	16	ineq2i	$$⊢B\cap \left\{y|\neg \phi \right\}=B\cap \left(\mathrm{V}\setminus C\right)$$
18	9 17	eqtr2i	$$⊢B\cap \left(\mathrm{V}\setminus C\right)=\left\{y\in B|\neg \phi \right\}$$
19	8 18	uneq12i	$$⊢\left(A\cap C\right)\cup \left(B\cap \left(\mathrm{V}\setminus C\right)\right)=\left\{y\in A|\phi \right\}\cup \left\{y\in B|\neg \phi \right\}$$
20	2 19	eqtr4i	$$⊢if\left(\phi ,A,B\right)=\left(A\cap C\right)\cup \left(B\cap \left(\mathrm{V}\setminus C\right)\right)$$