Theorem sb7h 2454

Description: This version of dfsb7 2455 does not require that ph 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-5 1839 i.e. that doesn't have the concept of a variable not occurring in a wff. (df-sb 1881 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	|- ( ph -> A. z ph )

Assertion

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

Distinct variable group:   y, z

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

Proof of Theorem sb7h

Step	Hyp	Ref	Expression
1		sb7h.1	. . 3 |- ( ph -> A. z ph )
2	1	nf5i 2024	. 2 |- F/ z ph
3	2	sb7f 2453	1 |- ( [ y / x ] ph <-> E. z ( z = y /\ E. x ( x = z /\ ph ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   E.wex 1704   [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by: (None)