Theorem tpeq3

Description: Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011)

	Ref	Expression
Assertion	tpeq3	$$⊢A=B\to \left\{C,D,A\right\}=\left\{C,D,B\right\}$$

Proof

Step	Hyp	Ref	Expression
1		sneq	$$⊢A=B\to \left\{A\right\}=\left\{B\right\}$$
2	1	uneq2d	$$⊢A=B\to \left\{C,D\right\}\cup \left\{A\right\}=\left\{C,D\right\}\cup \left\{B\right\}$$
3		df-tp	$$⊢\left\{C,D,A\right\}=\left\{C,D\right\}\cup \left\{A\right\}$$
4		df-tp	$$⊢\left\{C,D,B\right\}=\left\{C,D\right\}\cup \left\{B\right\}$$
5	2 3 4	3eqtr4g	$$⊢A=B\to \left\{C,D,A\right\}=\left\{C,D,B\right\}$$