Theorem sb7h 2121

Description: This version of dfsb7 2119 does not require that φ and z be distinct. This permits it to be used as a definition for substitution in a formalization that omits the logically redundant axiom ax-17 1616 i.e. that doesn't have the concept of a variable not occurring in a wff. (df-sb 1649 is also suitable, but its mixing of free and bound variables is distasteful to some logicians.) (Contributed by NM, 26-Jul-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Hypothesis

Ref	Expression
sb7h.1	⊢ (φ → ∀zφ)

Assertion

Ref	Expression
sb7h	⊢ ([y / x]φ ↔ ∃z(z = y ∧ ∃x(x = z ∧ φ)))

Distinct variable groups:   x,z   y,z

Allowed substitution hints:   φ(x,y,z)

Proof of Theorem sb7h

Step	Hyp	Ref	Expression
1		sb7h.1	. . 3 ⊢ (φ → ∀zφ)
2	1	nfi 1551	. 2 ⊢ Ⅎzφ
3	2	sb7f 2120	1 ⊢ ([y / x]φ ↔ ∃z(z = y ∧ ∃x(x = z ∧ φ)))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃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)