Theorem ifeq1

Description: Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004) (Revised by Mario Carneiro, 8-Sep-2013)

	Ref	Expression
Assertion	ifeq1	$$⊢A=B\to if\left(\phi ,A,C\right)=if\left(\phi ,B,C\right)$$

Proof

Step	Hyp	Ref	Expression
1		rabeq	$$⊢A=B\to \left\{x\in A|\phi \right\}=\left\{x\in B|\phi \right\}$$
2	1	uneq1d	$$⊢A=B\to \left\{x\in A|\phi \right\}\cup \left\{x\in C|\neg \phi \right\}=\left\{x\in B|\phi \right\}\cup \left\{x\in C|\neg \phi \right\}$$
3		dfif6	$$⊢if\left(\phi ,A,C\right)=\left\{x\in A|\phi \right\}\cup \left\{x\in C|\neg \phi \right\}$$
4		dfif6	$$⊢if\left(\phi ,B,C\right)=\left\{x\in B|\phi \right\}\cup \left\{x\in C|\neg \phi \right\}$$
5	2 3 4	3eqtr4g	$$⊢A=B\to if\left(\phi ,A,C\right)=if\left(\phi ,B,C\right)$$