Theorem spcimgft

Description: A closed version of spcimgf . (Contributed by Mario Carneiro, 4-Jan-2017)

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

Proof

Step	Hyp	Ref	Expression
1		spcimgft.1	$$⊢Ⅎx\psi$$
2		spcimgft.2	$$⊢\underset{\_}{Ⅎ}xA$$
3		elex	$$⊢A\in B\to A\in \mathrm{V}$$
4	2	issetf	$$⊢A\in \mathrm{V}\leftrightarrow \exists xx=A$$
5		exim	$$⊢\forall x\left(x=A\to \left(\phi \to \psi \right)\right)\to \left(\exists xx=A\to \exists x\left(\phi \to \psi \right)\right)$$
6	4 5	biimtrid	$$⊢\forall x\left(x=A\to \left(\phi \to \psi \right)\right)\to \left(A\in \mathrm{V}\to \exists x\left(\phi \to \psi \right)\right)$$
7	1	19.36	$$⊢\exists x\left(\phi \to \psi \right)\leftrightarrow \left(\forall x\phi \to \psi \right)$$
8	6 7	imbitrdi	$$⊢\forall x\left(x=A\to \left(\phi \to \psi \right)\right)\to \left(A\in \mathrm{V}\to \left(\forall x\phi \to \psi \right)\right)$$
9	3 8	syl5	$$⊢\forall x\left(x=A\to \left(\phi \to \psi \right)\right)\to \left(A\in B\to \left(\forall x\phi \to \psi \right)\right)$$