Theorem tpeq1

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

	Ref	Expression
Assertion	tpeq1	$$⊢A=B\to \left\{A,C,D\right\}=\left\{B,C,D\right\}$$

Proof

Step	Hyp	Ref	Expression
1		preq1	$$⊢A=B\to \left\{A,C\right\}=\left\{B,C\right\}$$
2	1	uneq1d	$$⊢A=B\to \left\{A,C\right\}\cup \left\{D\right\}=\left\{B,C\right\}\cup \left\{D\right\}$$
3		df-tp	$$⊢\left\{A,C,D\right\}=\left\{A,C\right\}\cup \left\{D\right\}$$
4		df-tp	$$⊢\left\{B,C,D\right\}=\left\{B,C\right\}\cup \left\{D\right\}$$
5	2 3 4	3eqtr4g	$$⊢A=B\to \left\{A,C,D\right\}=\left\{B,C,D\right\}$$