Theorem vtoclgft

Description: Closed theorem form of vtoclgf . The reverse implication is proven in ceqsal1t . See ceqsalt for a version with x and A disjoint. (Contributed by NM, 17-Feb-2013) (Revised by Mario Carneiro, 12-Oct-2016) (Proof shortened by JJ, 11-Aug-2021) Avoid ax-13 . (Revised by GG, 6-Oct-2023)

	Ref	Expression
Assertion	vtoclgft	$$⊢\left(\left(\underset{\_}{Ⅎ}xA\wedge Ⅎx\psi \right)\wedge \left(\forall x\left(x=A\to \left(\phi \leftrightarrow \psi \right)\right)\wedge \forall x\phi \right)\wedge A\in V\right)\to \psi$$

Proof

Step	Hyp	Ref	Expression
1		biimp	$$⊢\left(\phi \leftrightarrow \psi \right)\to \left(\phi \to \psi \right)$$
2	1	imim2i	$$⊢\left(x=A\to \left(\phi \leftrightarrow \psi \right)\right)\to \left(x=A\to \left(\phi \to \psi \right)\right)$$
3	2	alimi	$$⊢\forall x\left(x=A\to \left(\phi \leftrightarrow \psi \right)\right)\to \forall x\left(x=A\to \left(\phi \to \psi \right)\right)$$
4		spcimgft	$$⊢\left(\left(\underset{\_}{Ⅎ}xA\wedge Ⅎx\psi \right)\wedge \forall x\left(x=A\to \left(\phi \to \psi \right)\right)\right)\to \left(A\in V\to \left(\forall x\phi \to \psi \right)\right)$$
5	3 4	sylan2	$$⊢\left(\left(\underset{\_}{Ⅎ}xA\wedge Ⅎx\psi \right)\wedge \forall x\left(x=A\to \left(\phi \leftrightarrow \psi \right)\right)\right)\to \left(A\in V\to \left(\forall x\phi \to \psi \right)\right)$$
6	5	com23	$$⊢\left(\left(\underset{\_}{Ⅎ}xA\wedge Ⅎx\psi \right)\wedge \forall x\left(x=A\to \left(\phi \leftrightarrow \psi \right)\right)\right)\to \left(\forall x\phi \to \left(A\in V\to \psi \right)\right)$$
7	6	impr	$$⊢\left(\left(\underset{\_}{Ⅎ}xA\wedge Ⅎx\psi \right)\wedge \left(\forall x\left(x=A\to \left(\phi \leftrightarrow \psi \right)\right)\wedge \forall x\phi \right)\right)\to \left(A\in V\to \psi \right)$$
8	7	3impia	$$⊢\left(\left(\underset{\_}{Ⅎ}xA\wedge Ⅎx\psi \right)\wedge \left(\forall x\left(x=A\to \left(\phi \leftrightarrow \psi \right)\right)\wedge \forall x\phi \right)\wedge A\in V\right)\to \psi$$