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Metamath Proof Explorer

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 ) )