Theorem bj-sblem2

Description: Lemma for substitution. (Contributed by BJ, 23-Jul-2023)

	Ref	Expression
Assertion	bj-sblem2	$$⊢\forall x\left(\phi \to \left(\chi \to \psi \right)\right)\to \left(\left(\exists x\phi \to \chi \right)\to \forall x\left(\phi \to \psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		19.23v	$$⊢\forall x\left(\phi \to \chi \right)\leftrightarrow \left(\exists x\phi \to \chi \right)$$
2		ax-2	$$⊢\left(\phi \to \left(\chi \to \psi \right)\right)\to \left(\left(\phi \to \chi \right)\to \left(\phi \to \psi \right)\right)$$
3	2	al2imi	$$⊢\forall x\left(\phi \to \left(\chi \to \psi \right)\right)\to \left(\forall x\left(\phi \to \chi \right)\to \forall x\left(\phi \to \psi \right)\right)$$
4	1 3	biimtrrid	$$⊢\forall x\left(\phi \to \left(\chi \to \psi \right)\right)\to \left(\left(\exists x\phi \to \chi \right)\to \forall x\left(\phi \to \psi \right)\right)$$