Theorem axc16g

Description: Generalization of axc16 . Use the latter when sufficient. This proof only requires, on top of { ax-1 -- ax-7 }, theorem ax12v . (Contributed by NM, 15-May-1993) (Proof shortened by Andrew Salmon, 25-May-2011) (Proof shortened by Wolf Lammen, 18-Feb-2018) Remove dependency on ax-13 , along an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019) (Revised by BJ, 7-Jul-2021) Shorten axc11rv . (Revised by Wolf Lammen, 11-Oct-2021)

	Ref	Expression
Assertion	axc16g	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( 𝜑 → ∀ 𝑧 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		aevlem	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ∀ 𝑧 𝑧 = 𝑤 )
2		ax12v	⊢ ( 𝑧 = 𝑤 → ( 𝜑 → ∀ 𝑧 ( 𝑧 = 𝑤 → 𝜑 ) ) )
3	2	sps	⊢ ( ∀ 𝑧 𝑧 = 𝑤 → ( 𝜑 → ∀ 𝑧 ( 𝑧 = 𝑤 → 𝜑 ) ) )
4		pm2.27	⊢ ( 𝑧 = 𝑤 → ( ( 𝑧 = 𝑤 → 𝜑 ) → 𝜑 ) )
5	4	al2imi	⊢ ( ∀ 𝑧 𝑧 = 𝑤 → ( ∀ 𝑧 ( 𝑧 = 𝑤 → 𝜑 ) → ∀ 𝑧 𝜑 ) )
6	3 5	syld	⊢ ( ∀ 𝑧 𝑧 = 𝑤 → ( 𝜑 → ∀ 𝑧 𝜑 ) )
7	1 6	syl	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( 𝜑 → ∀ 𝑧 𝜑 ) )