Theorem ax11b 1995

Description: A bidirectional version of ax11o 1994. (Contributed by NM, 30-Jun-2006.)

Assertion

Ref	Expression
ax11b	⊢ ((¬ ∀x x = y ∧ x = y) → (φ ↔ ∀x(x = y → φ)))

Proof of Theorem ax11b

Step	Hyp	Ref	Expression
1		ax11o 1994	. . 3 ⊢ (¬ ∀x x = y → (x = y → (φ → ∀x(x = y → φ))))
2	1	imp 418	. 2 ⊢ ((¬ ∀x x = y ∧ x = y) → (φ → ∀x(x = y → φ)))
3		sp 1747	. . . 4 ⊢ (∀x(x = y → φ) → (x = y → φ))
4	3	com12 27	. . 3 ⊢ (x = y → (∀x(x = y → φ) → φ))
5	4	adantl 452	. 2 ⊢ ((¬ ∀x x = y ∧ x = y) → (∀x(x = y → φ) → φ))
6	2, 5	impbid 183	1 ⊢ ((¬ ∀x x = y ∧ x = y) → (φ ↔ ∀x(x = y → φ)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by: (None)