Theorem dfsb3 2056

Description: An alternate definition of proper substitution df-sb 1649 that uses only primitive connectives (no defined terms) on the right-hand side. (Contributed by NM, 6-Mar-2007.)

Assertion

Ref	Expression
dfsb3	⊢ ([y / x]φ ↔ ((x = y → ¬ φ) → ∀x(x = y → φ)))

Proof of Theorem dfsb3

Step	Hyp	Ref	Expression
1		df-or 359	. 2 ⊢ (((x = y ∧ φ) ∨ ∀x(x = y → φ)) ↔ (¬ (x = y ∧ φ) → ∀x(x = y → φ)))
2		dfsb2 2055	. 2 ⊢ ([y / x]φ ↔ ((x = y ∧ φ) ∨ ∀x(x = y → φ)))
3		imnan 411	. . 3 ⊢ ((x = y → ¬ φ) ↔ ¬ (x = y ∧ φ))
4	3	imbi1i 315	. 2 ⊢ (((x = y → ¬ φ) → ∀x(x = y → φ)) ↔ (¬ (x = y ∧ φ) → ∀x(x = y → φ)))
5	1, 2, 4	3bitr4i 268	1 ⊢ ([y / x]φ ↔ ((x = y → ¬ φ) → ∀x(x = y → φ)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358  ∀wal 1540  [wsb 1648

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-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649

This theorem is referenced by: (None)