Theorem axc16g-o

Description: A generalization of Axiom ax-c16 . Version of axc16g using ax-c11 . (Contributed by NM, 15-May-1993) (Proof shortened by Andrew Salmon, 25-May-2011) (Proof modification is discouraged.) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		aev-o	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ∀ 𝑧 𝑧 = 𝑥 )
2		ax-c16	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( 𝜑 → ∀ 𝑥 𝜑 ) )
3		biidd	⊢ ( ∀ 𝑧 𝑧 = 𝑥 → ( 𝜑 ↔ 𝜑 ) )
4	3	dral1-o	⊢ ( ∀ 𝑧 𝑧 = 𝑥 → ( ∀ 𝑧 𝜑 ↔ ∀ 𝑥 𝜑 ) )
5	4	biimprd	⊢ ( ∀ 𝑧 𝑧 = 𝑥 → ( ∀ 𝑥 𝜑 → ∀ 𝑧 𝜑 ) )
6	1 2 5	sylsyld	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( 𝜑 → ∀ 𝑧 𝜑 ) )