Theorem dfsb7 2119

Description: An alternate definition of proper substitution df-sb 1649. By introducing a dummy variable z in the definiens, we are able to eliminate any distinct variable restrictions among the variables x, y, and φ of the definiendum. No distinct variable conflicts arise because z effectively insulates x from y. To achieve this, we use a chain of two substitutions in the form of sb5 2100, first z for x then y for z. Compare Definition 2.1'' of [Quine] p. 17, which is obtained from this theorem by applying df-clab 2340. Theorem sb7h 2121 provides a version where φ and z don't have to be distinct. (Contributed by NM, 28-Jan-2004.)

Assertion

Ref	Expression
dfsb7	⊢ ([y / x]φ ↔ ∃z(z = y ∧ ∃x(x = z ∧ φ)))

Distinct variable groups:   x,z   y,z   φ,z

Allowed substitution hints:   φ(x,y)

Proof of Theorem dfsb7

Step	Hyp	Ref	Expression
1		sb5 2100	. . 3 ⊢ ([z / x]φ ↔ ∃x(x = z ∧ φ))
2	1	sbbii 1653	. 2 ⊢ ([y / z][z / x]φ ↔ [y / z]∃x(x = z ∧ φ))
3		nfv 1619	. . 3 ⊢ Ⅎzφ
4	3	sbco2 2086	. 2 ⊢ ([y / z][z / x]φ ↔ [y / x]φ)
5		sb5 2100	. 2 ⊢ ([y / z]∃x(x = z ∧ φ) ↔ ∃z(z = y ∧ ∃x(x = z ∧ φ)))
6	2, 4, 5	3bitr3i 266	1 ⊢ ([y / x]φ ↔ ∃z(z = y ∧ ∃x(x = z ∧ φ)))

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541  [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)