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Metamath Proof Explorer

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	$⊢ φ \to (ψ \leftrightarrow χ)$
	bj-ax12i.2	$⊢ χ \to \forall x χ$
Assertion	bj-ax12i	$⊢ φ \to (ψ \to \forall x (φ \to ψ))$

Proof

Step	Hyp	Ref	Expression
1		bj-ax12i.1	$⊢ φ \to (ψ \leftrightarrow χ)$
2		bj-ax12i.2	$⊢ χ \to \forall x χ$
3	2	a1i	$⊢ φ \to (χ \to \forall x χ)$
4	1 3	bj-ax12ig	$⊢ φ \to (ψ \to \forall x (φ \to ψ))$