Theorem ax12ev2

Description: Version of ax12v2 rewritten to use an existential quantifier. One direction of sbalex without the universal quantifier, avoiding ax-10 . (Contributed by SN, 14-Aug-2025)

	Ref	Expression
Assertion	ax12ev2	⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) → ( 𝑥 = 𝑦 → 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		exnalimn	⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ↔ ¬ ∀ 𝑥 ( 𝑥 = 𝑦 → ¬ 𝜑 ) )
2		ax12v2	⊢ ( 𝑥 = 𝑦 → ( ¬ 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → ¬ 𝜑 ) ) )
3	2	con1d	⊢ ( 𝑥 = 𝑦 → ( ¬ ∀ 𝑥 ( 𝑥 = 𝑦 → ¬ 𝜑 ) → 𝜑 ) )
4	1 3	biimtrid	⊢ ( 𝑥 = 𝑦 → ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) → 𝜑 ) )
5	4	com12	⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) → ( 𝑥 = 𝑦 → 𝜑 ) )