Theorem gencl

Description: Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996)

	Ref	Expression
Hypotheses	gencl.1	$$⊢\theta \leftrightarrow \exists x\left(\chi \wedge A=B\right)$$
	gencl.2	$$⊢A=B\to \left(\phi \leftrightarrow \psi \right)$$
	gencl.3	$$⊢\chi \to \phi$$
Assertion	gencl	$$⊢\theta \to \psi$$

Proof

Step	Hyp	Ref	Expression
1		gencl.1	$$⊢\theta \leftrightarrow \exists x\left(\chi \wedge A=B\right)$$
2		gencl.2	$$⊢A=B\to \left(\phi \leftrightarrow \psi \right)$$
3		gencl.3	$$⊢\chi \to \phi$$
4	3 2	imbitrid	$$⊢A=B\to \left(\chi \to \psi \right)$$
5	4	impcom	$$⊢\left(\chi \wedge A=B\right)\to \psi$$
6	5	exlimiv	$$⊢\exists x\left(\chi \wedge A=B\right)\to \psi$$
7	1 6	sylbi	$$⊢\theta \to \psi$$