Theorem sb7f 2453

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.) (Revised by Mario Carneiro, 6-Oct-2016.)

Hypothesis

Ref	Expression
sb7f.1	|- F/ z ph

Assertion

Ref	Expression
sb7f	|- ( [ 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 sb7f

Step	Hyp	Ref	Expression
1		sb7f.1	. . . 4 |- F/ z ph
2	1	sb5f 2386	. . 3 |- ( [ z / x ] ph <-> E. x ( x = z /\ ph ) )
3	2	sbbii 1887	. 2 |- ( [ y / z ] [ z / x ] ph <-> [ y / z ] E. x ( x = z /\ ph ) )
4	1	sbco2 2415	. 2 |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )
5		sb5 2430	. 2 |- ( [ y / z ] E. x ( x = z /\ ph ) <-> E. z ( z = y /\ E. x ( x = z /\ ph ) ) )
6	3, 4, 5	3bitr3i 290	1 |- ( [ y / x ] ph <-> E. z ( z = y /\ E. x ( x = z /\ ph ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   E.wex 1704   F/wnf 1708   [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:  sb7h  2454  dfsb7  2455