Theorem cbvopabv

Description: Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 15-Oct-1996)

	Ref	Expression
Hypothesis	cbvopabv.1	$$⊢\left(x=z\wedge y=w\right)\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	cbvopabv	$$⊢\left\{⟨x,y⟩|\phi \right\}=\left\{⟨z,w⟩|\psi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		cbvopabv.1	$$⊢\left(x=z\wedge y=w\right)\to \left(\phi \leftrightarrow \psi \right)$$
2		nfv	$$⊢Ⅎz\phi$$
3		nfv	$$⊢Ⅎw\phi$$
4		nfv	$$⊢Ⅎx\psi$$
5		nfv	$$⊢Ⅎy\psi$$
6	2 3 4 5 1	cbvopab	$$⊢\left\{⟨x,y⟩|\phi \right\}=\left\{⟨z,w⟩|\psi \right\}$$