Theorem sb7h 2121

Description: This version of dfsb7 2119 does not require that ph and z be disjoint. 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 does not have the concept of a variable not occurring in a formula. (Contributed by NM, 26-Jul-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Hypothesis

Ref	Expression
sb7h.1	|- (ph -> A.zph)

Assertion

Ref	Expression
sb7h	|- ([y / x]ph <-> E.z(z = y /\ E.x(x = z /\ ph)))

Distinct variable groups:   x,z   y,z

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

Proof of Theorem sb7h

Step	Hyp	Ref	Expression
1		sb7h.1	. . 3 |- (ph -> A.zph)
2	1	nfi 1551	. 2 |- F/zph
3	2	sb7f 2120	1 |- ([y / x]ph <-> E.z(z = y /\ E.x(x = z /\ ph)))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540  E.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)