Theorem ax12b 2446

Description: A bidirectional version of axc15 2444. Usage of this theorem is discouraged because it depends on ax-13 2390. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)

Assertion

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

Proof of Theorem ax12b

Step	Hyp	Ref	Expression
1		axc15 2444	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
2	1	imp 409	. 2 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑥 = 𝑦) → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
3		sp 2182	. . . 4 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → (𝑥 = 𝑦 → 𝜑))
4	3	com12 32	. . 3 ⊢ (𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦 → 𝜑) → 𝜑))
5	4	adantl 484	. 2 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑥 = 𝑦) → (∀𝑥(𝑥 = 𝑦 → 𝜑) → 𝜑))
6	2, 5	impbid 214	1 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑥 = 𝑦) → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-12 2177  ax-13 2390

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-nf 1785

This theorem is referenced by: (None)