Theorem bj-nnfim1

Description: A consequence of nonfreeness in the antecedent and the consequent of an implication. (Contributed by BJ, 27-Aug-2023)

	Ref	Expression
Assertion	bj-nnfim1	$$⊢\left(Ⅎ\text{'}x\phi \wedge Ⅎ\text{'}x\psi \right)\to \left(\left(\phi \to \psi \right)\to \left(\exists x\phi \to \forall x\psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		bj-nnfe	$$⊢Ⅎ\text{'}x\phi \to \left(\exists x\phi \to \phi \right)$$
2		bj-nnfa	$$⊢Ⅎ\text{'}x\psi \to \left(\psi \to \forall x\psi \right)$$
3		imim12	$$⊢\left(\exists x\phi \to \phi \right)\to \left(\left(\psi \to \forall x\psi \right)\to \left(\left(\phi \to \psi \right)\to \left(\exists x\phi \to \forall x\psi \right)\right)\right)$$
4	3	imp	$$⊢\left(\left(\exists x\phi \to \phi \right)\wedge \left(\psi \to \forall x\psi \right)\right)\to \left(\left(\phi \to \psi \right)\to \left(\exists x\phi \to \forall x\psi \right)\right)$$
5	1 2 4	syl2an	$$⊢\left(Ⅎ\text{'}x\phi \wedge Ⅎ\text{'}x\psi \right)\to \left(\left(\phi \to \psi \right)\to \left(\exists x\phi \to \forall x\psi \right)\right)$$