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Metamath Proof Explorer

Theorem hbae-o

Description: All variables are effectively bound in an identical variable specifier. Version of hbae using ax-c11 . (Contributed by NM, 13-May-1993) (Proof modification is discouraged.) (New usage is discouraged.)

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

Proof

Step Hyp Ref Expression
1 ax-c5 ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦 )
2 ax-c9 ⊢ ( ¬ ∀ 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 ax-c11 ⊢ ( ∀ 𝑥 𝑥 = 𝑧 → ( ∀ 𝑥 𝑥 = 𝑦 → ∀ 𝑧 𝑥 = 𝑦 ) )
5 4 aecoms-o ⊢ ( ∀ 𝑧 𝑧 = 𝑥 → ( ∀ 𝑥 𝑥 = 𝑦 → ∀ 𝑧 𝑥 = 𝑦 ) )
6 ax-c11 ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 𝑥 = 𝑦 → ∀ 𝑦 𝑥 = 𝑦 ) )
7 6 pm2.43i ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ∀ 𝑦 𝑥 = 𝑦 )
8 ax-c11 ⊢ ( ∀ 𝑦 𝑦 = 𝑧 → ( ∀ 𝑦 𝑥 = 𝑦 → ∀ 𝑧 𝑥 = 𝑦 ) )
9 7 8 syl5 ⊢ ( ∀ 𝑦 𝑦 = 𝑧 → ( ∀ 𝑥 𝑥 = 𝑦 → ∀ 𝑧 𝑥 = 𝑦 ) )
10 9 aecoms-o ⊢ ( ∀ 𝑧 𝑧 = 𝑦 → ( ∀ 𝑥 𝑥 = 𝑦 → ∀ 𝑧 𝑥 = 𝑦 ) )
11 3 5 10 pm2.61ii ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ∀ 𝑧 𝑥 = 𝑦 )
12 11 axc4i-o ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ∀ 𝑥 ∀ 𝑧 𝑥 = 𝑦 )
13 ax-11 ⊢ ( ∀ 𝑥 ∀ 𝑧 𝑥 = 𝑦 → ∀ 𝑧 ∀ 𝑥 𝑥 = 𝑦 )
14 12 13 syl ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ∀ 𝑧 ∀ 𝑥 𝑥 = 𝑦 )