Theorem ceqsal1t

Description: One direction of ceqsalt is based on fewer assumptions and fewer axioms. It is at the same time the reverse direction of vtoclgft . Extracted from a proof of ceqsalt . (Contributed by Wolf Lammen, 25-Mar-2025)

	Ref	Expression
Assertion	ceqsal1t	$$⊢\left(Ⅎx\psi \wedge \forall x\left(x=A\to \left(\phi \leftrightarrow \psi \right)\right)\right)\to \left(\psi \to \forall x\left(x=A\to \phi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		biimpr	$$⊢\left(\phi \leftrightarrow \psi \right)\to \left(\psi \to \phi \right)$$
2	1	imim2i	$$⊢\left(x=A\to \left(\phi \leftrightarrow \psi \right)\right)\to \left(x=A\to \left(\psi \to \phi \right)\right)$$
3	2	com23	$$⊢\left(x=A\to \left(\phi \leftrightarrow \psi \right)\right)\to \left(\psi \to \left(x=A\to \phi \right)\right)$$
4	3	alimi	$$⊢\forall x\left(x=A\to \left(\phi \leftrightarrow \psi \right)\right)\to \forall x\left(\psi \to \left(x=A\to \phi \right)\right)$$
5		19.21t	$$⊢Ⅎx\psi \to \left(\forall x\left(\psi \to \left(x=A\to \phi \right)\right)\leftrightarrow \left(\psi \to \forall x\left(x=A\to \phi \right)\right)\right)$$
6	4 5	imbitrid	$$⊢Ⅎx\psi \to \left(\forall x\left(x=A\to \left(\phi \leftrightarrow \psi \right)\right)\to \left(\psi \to \forall x\left(x=A\to \phi \right)\right)\right)$$
7	6	imp	$$⊢\left(Ⅎx\psi \wedge \forall x\left(x=A\to \left(\phi \leftrightarrow \psi \right)\right)\right)\to \left(\psi \to \forall x\left(x=A\to \phi \right)\right)$$