Theorem ax12b

Description: A bidirectional version of axc15 . Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 30-Jun-2006) (New usage is discouraged.)

	Ref	Expression
Assertion	ax12b	⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ 𝑥 = 𝑦 ) → ( 𝜑 ↔ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) )

Proof

Step	Hyp	Ref	Expression
1		axc15	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( 𝑥 = 𝑦 → ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ) )
2	1	imp	⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ 𝑥 = 𝑦 ) → ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) )
3		sp	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → ( 𝑥 = 𝑦 → 𝜑 ) )
4	3	com12	⊢ ( 𝑥 = 𝑦 → ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → 𝜑 ) )
5	4	adantl	⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ 𝑥 = 𝑦 ) → ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → 𝜑 ) )
6	2 5	impbid	⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ 𝑥 = 𝑦 ) → ( 𝜑 ↔ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) )