Theorem spcgf

Description: Rule of specialization, using implicit substitution. Compare Theorem 7.3 of Quine p. 44. (Contributed by NM, 2-Feb-1997) (Revised by Andrew Salmon, 12-Aug-2011)

	Ref	Expression
Hypotheses	spcgf.1	$$⊢\underset{\_}{Ⅎ}xA$$
	spcgf.2	$$⊢Ⅎx\psi$$
	spcgf.3	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	spcgf	$$⊢A\in V\to \left(\forall x\phi \to \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		spcgf.1	$$⊢\underset{\_}{Ⅎ}xA$$
2		spcgf.2	$$⊢Ⅎx\psi$$
3		spcgf.3	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
4	2 1	spcgft	$$⊢\forall x\left(x=A\to \left(\phi \leftrightarrow \psi \right)\right)\to \left(A\in V\to \left(\forall x\phi \to \psi \right)\right)$$
5	4 3	mpg	$$⊢A\in V\to \left(\forall x\phi \to \psi \right)$$