Theorem preq12d

Description: Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012)

	Ref	Expression
Hypotheses	preq1d.1	$$⊢\phi \to A=B$$
	preq12d.2	$$⊢\phi \to C=D$$
Assertion	preq12d	$$⊢\phi \to \left\{A,C\right\}=\left\{B,D\right\}$$

Proof

Step	Hyp	Ref	Expression
1		preq1d.1	$$⊢\phi \to A=B$$
2		preq12d.2	$$⊢\phi \to C=D$$
3		preq12	$$⊢\left(A=B\wedge C=D\right)\to \left\{A,C\right\}=\left\{B,D\right\}$$
4	1 2 3	syl2anc	$$⊢\phi \to \left\{A,C\right\}=\left\{B,D\right\}$$