Theorem wl-speqv

Description: Under the assumption -. x = y a specialized version of sp is provable from Tarski's FOL and ax13v only. Note that this reverts the implication in ax13lem1 , so in fact ( -. x = y -> ( A. x z = y <-> z = y ) ) holds. (Contributed by Wolf Lammen, 17-Apr-2021)

	Ref	Expression
Assertion	wl-speqv	⊢ ( ¬ 𝑥 = 𝑦 → ( ∀ 𝑥 𝑧 = 𝑦 → 𝑧 = 𝑦 ) )

Proof

Step	Hyp	Ref	Expression
1		19.2	⊢ ( ∀ 𝑥 𝑧 = 𝑦 → ∃ 𝑥 𝑧 = 𝑦 )
2		ax13lem2	⊢ ( ¬ 𝑥 = 𝑦 → ( ∃ 𝑥 𝑧 = 𝑦 → 𝑧 = 𝑦 ) )
3	1 2	syl5	⊢ ( ¬ 𝑥 = 𝑦 → ( ∀ 𝑥 𝑧 = 𝑦 → 𝑧 = 𝑦 ) )