Theorem bj-ax12w

Description: The general statement that ax12w proves. (Contributed by BJ, 20-Mar-2020)

	Ref	Expression
Hypotheses	bj-ax12w.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
	bj-ax12w.2	$$⊢y=z\to \left(\psi \leftrightarrow \theta \right)$$
Assertion	bj-ax12w	$$⊢\phi \to \left(\forall y\psi \to \forall x\left(\phi \to \psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		bj-ax12w.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
2		bj-ax12w.2	$$⊢y=z\to \left(\psi \leftrightarrow \theta \right)$$
3	2	spw	$$⊢\forall y\psi \to \psi$$
4	1	bj-ax12wlem	$$⊢\phi \to \left(\psi \to \forall x\left(\phi \to \psi \right)\right)$$
5	3 4	syl5	$$⊢\phi \to \left(\forall y\psi \to \forall x\left(\phi \to \psi \right)\right)$$