Theorem bj-ax12ig

Description: A lemma used to prove a weak form of the axiom of substitution. A generalization of bj-ax12i . (Contributed by BJ, 19-Dec-2020)

	Ref	Expression
Hypotheses	bj-ax12ig.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
	bj-ax12ig.2	$$⊢\phi \to \left(\chi \to \forall x\chi \right)$$
Assertion	bj-ax12ig	$$⊢\phi \to \left(\psi \to \forall x\left(\phi \to \psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		bj-ax12ig.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
2		bj-ax12ig.2	$$⊢\phi \to \left(\chi \to \forall x\chi \right)$$
3	1	pm5.32i	$$⊢\left(\phi \wedge \psi \right)\leftrightarrow \left(\phi \wedge \chi \right)$$
4	2	imp	$$⊢\left(\phi \wedge \chi \right)\to \forall x\chi$$
5	1	biimprcd	$$⊢\chi \to \left(\phi \to \psi \right)$$
6	4 5	sylg	$$⊢\left(\phi \wedge \chi \right)\to \forall x\left(\phi \to \psi \right)$$
7	3 6	sylbi	$$⊢\left(\phi \wedge \psi \right)\to \forall x\left(\phi \to \psi \right)$$
8	7	ex	$$⊢\phi \to \left(\psi \to \forall x\left(\phi \to \psi \right)\right)$$