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 → ( 𝜑 ↔ 𝜓 ) )
Assertion	ax12wlem	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ax12wlemw.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
2		ax-5	⊢ ( 𝜓 → ∀ 𝑥 𝜓 )
3	1 2	ax12i	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) )