Theorem axc16gALT

Description: Alternate proof of axc16g that uses df-sb and requires ax-10 , ax-11 , ax-13 . (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	axc16gALT	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( 𝜑 → ∀ 𝑧 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		aev	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ∀ 𝑧 𝑧 = 𝑥 )
2		axc16ALT	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( 𝜑 → ∀ 𝑥 𝜑 ) )
3		biidd	⊢ ( ∀ 𝑧 𝑧 = 𝑥 → ( 𝜑 ↔ 𝜑 ) )
4	3	dral1	⊢ ( ∀ 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 ) )