Theorem equs5aALT

Description: Alternate proof of equs5a . Uses ax-12 but not ax-13 . (Contributed by NM, 2-Feb-2007) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	equs5aALT	⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ ∀ 𝑦 𝜑 ) → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		nfa1	⊢ Ⅎ 𝑥 ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 )
2		ax-12	⊢ ( 𝑥 = 𝑦 → ( ∀ 𝑦 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) )
3	2	imp	⊢ ( ( 𝑥 = 𝑦 ∧ ∀ 𝑦 𝜑 ) → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) )
4	1 3	exlimi	⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ ∀ 𝑦 𝜑 ) → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) )