Theorem bj-ax12i

Description: A weakening of bj-ax12ig that is sufficient to prove a weak form of the axiom of substitution ax-12 . The general statement of which ax12i is an instance. (Contributed by BJ, 29-Sep-2019)

	Ref	Expression
Hypotheses	bj-ax12i.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
	bj-ax12i.2	$$⊢\chi \to \forall x\chi$$
Assertion	bj-ax12i	$$⊢\phi \to \left(\psi \to \forall x\left(\phi \to \psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		bj-ax12i.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
2		bj-ax12i.2	$$⊢\chi \to \forall x\chi$$
3	2	a1i	$$⊢\phi \to \left(\chi \to \forall x\chi \right)$$
4	1 3	bj-ax12ig	$$⊢\phi \to \left(\psi \to \forall x\left(\phi \to \psi \right)\right)$$