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Metamath Proof Explorer

Theorem hbae

Description: All variables are effectively bound in an identical variable specifier. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker hbaev when possible. (Contributed by NM, 13-May-1993) (Proof shortened by Wolf Lammen, 21-Apr-2018) (New usage is discouraged.)

Ref Expression
Assertion hbae ( ∀ 𝑥 𝑥 = 𝑦 → ∀ 𝑧 ∀ 𝑥 𝑥 = 𝑦 )

Proof

Step Hyp Ref Expression
1 sp ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦 )
2 axc9 ⊢ ( ¬ ∀ 𝑧 𝑧 = 𝑥 → ( ¬ ∀ 𝑧 𝑧 = 𝑦 → ( 𝑥 = 𝑦 → ∀ 𝑧 𝑥 = 𝑦 ) ) )
3 1 2 syl7 ⊢ ( ¬ ∀ 𝑧 𝑧 = 𝑥 → ( ¬ ∀ 𝑧 𝑧 = 𝑦 → ( ∀ 𝑥 𝑥 = 𝑦 → ∀ 𝑧 𝑥 = 𝑦 ) ) )
4 axc11r ⊢ ( ∀ 𝑧 𝑧 = 𝑥 → ( ∀ 𝑥 𝑥 = 𝑦 → ∀ 𝑧 𝑥 = 𝑦 ) )
5 axc11 ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 𝑥 = 𝑦 → ∀ 𝑦 𝑥 = 𝑦 ) )
6 5 pm2.43i ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ∀ 𝑦 𝑥 = 𝑦 )
7 axc11r ⊢ ( ∀ 𝑧 𝑧 = 𝑦 → ( ∀ 𝑦 𝑥 = 𝑦 → ∀ 𝑧 𝑥 = 𝑦 ) )
8 6 7 syl5 ⊢ ( ∀ 𝑧 𝑧 = 𝑦 → ( ∀ 𝑥 𝑥 = 𝑦 → ∀ 𝑧 𝑥 = 𝑦 ) )
9 3 4 8 pm2.61ii ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ∀ 𝑧 𝑥 = 𝑦 )
10 9 axc4i ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ∀ 𝑥 ∀ 𝑧 𝑥 = 𝑦 )
11 ax-11 ⊢ ( ∀ 𝑥 ∀ 𝑧 𝑥 = 𝑦 → ∀ 𝑧 ∀ 𝑥 𝑥 = 𝑦 )
12 10 11 syl ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ∀ 𝑧 ∀ 𝑥 𝑥 = 𝑦 )