Theorem ax12b 2459

Description: A bidirectional version of axc15 2457. Usage of this theorem is discouraged because it depends on ax-13 2407. (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 2457	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
2	1	imp 412	. 2 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑥 = 𝑦) → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
3		sp 2222	. . . 4 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → (𝑥 = 𝑦 → 𝜑))
4	3	com12 33	. . 3 ⊢ (𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦 → 𝜑) → 𝜑))
5	4	adantl 487	. 2 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑥 = 𝑦) → (∀𝑥(𝑥 = 𝑦 → 𝜑) → 𝜑))
6	2, 5	impbid 215	1 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑥 = 𝑦) → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-12 2216  ax-13 2407

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817

This theorem is used by: (None)