Theorem spcgft

Description: A closed version of spcgf . (Contributed by Andrew Salmon, 6-Jun-2011) (Revised by Mario Carneiro, 4-Jan-2017)

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

Proof

Step	Hyp	Ref	Expression
1		spcimgft.1	$$⊢Ⅎx\psi$$
2		spcimgft.2	$$⊢\underset{\_}{Ⅎ}xA$$
3		biimp	$$⊢\left(\phi \leftrightarrow \psi \right)\to \left(\phi \to \psi \right)$$
4	3	imim2i	$$⊢\left(x=A\to \left(\phi \leftrightarrow \psi \right)\right)\to \left(x=A\to \left(\phi \to \psi \right)\right)$$
5	4	alimi	$$⊢\forall x\left(x=A\to \left(\phi \leftrightarrow \psi \right)\right)\to \forall x\left(x=A\to \left(\phi \to \psi \right)\right)$$
6	1 2	spcimgft	$$⊢\forall x\left(x=A\to \left(\phi \to \psi \right)\right)\to \left(A\in B\to \left(\forall x\phi \to \psi \right)\right)$$
7	5 6	syl	$$⊢\forall x\left(x=A\to \left(\phi \leftrightarrow \psi \right)\right)\to \left(A\in B\to \left(\forall x\phi \to \psi \right)\right)$$