This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem ax12wlem

Description: Lemma for weak version of ax-12 . Uses only Tarski's FOL axiom schemes. In some cases, this lemma may lead to shorter proofs than ax12w . (Contributed by NM, 10-Apr-2017)

		Ref	Expression
	Hypothesis	ax12wlemw.1	$⊢ x = y \to (φ \leftrightarrow ψ)$
	Assertion	ax12wlem	$⊢ x = y \to (φ \to \forall x (x = y \to φ))$

Proof

Step	Hyp	Ref	Expression
1		ax12wlemw.1	$⊢ x = y \to (φ \leftrightarrow ψ)$
2		ax-5	$⊢ ψ \to \forall x ψ$
3	1 2	ax12i	$⊢ x = y \to (φ \to \forall x (x = y \to φ))$