Theorem cbvopab1v

Description: Rule used to change the first bound variable in an ordered pair abstraction, using implicit substitution. (Contributed by NM, 31-Jul-2003) (Proof shortened by Eric Schmidt, 4-Apr-2007)

	Ref	Expression
Hypothesis	cbvopab1v.1	$$⊢x=z\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	cbvopab1v	$$⊢\left\{⟨x,y⟩|\phi \right\}=\left\{⟨z,y⟩|\psi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		cbvopab1v.1	$$⊢x=z\to \left(\phi \leftrightarrow \psi \right)$$
2		nfv	$$⊢Ⅎz\phi$$
3		nfv	$$⊢Ⅎx\psi$$
4	2 3 1	cbvopab1	$$⊢\left\{⟨x,y⟩|\phi \right\}=\left\{⟨z,y⟩|\psi \right\}$$