Theorem preq1

Description: Equality theorem for unordered pairs. (Contributed by NM, 29-Mar-1998)

	Ref	Expression
Assertion	preq1	$$⊢A=B\to \left\{A,C\right\}=\left\{B,C\right\}$$

Proof

Step	Hyp	Ref	Expression
1		sneq	$$⊢A=B\to \left\{A\right\}=\left\{B\right\}$$
2	1	uneq1d	$$⊢A=B\to \left\{A\right\}\cup \left\{C\right\}=\left\{B\right\}\cup \left\{C\right\}$$
3		df-pr	$$⊢\left\{A,C\right\}=\left\{A\right\}\cup \left\{C\right\}$$
4		df-pr	$$⊢\left\{B,C\right\}=\left\{B\right\}\cup \left\{C\right\}$$
5	2 3 4	3eqtr4g	$$⊢A=B\to \left\{A,C\right\}=\left\{B,C\right\}$$