Theorem sb7f

Description: This version of dfsb7 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-5 , i.e., that does not have the concept of a variable not occurring in a formula. (Definition dfsb1 is also suitable, but its mixing of free and bound variables is distasteful to some logicians.) Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 26-Jul-2006) (Revised by Mario Carneiro, 6-Oct-2016) (New usage is discouraged.)

	Ref	Expression
Hypothesis	sb7f.1	|- F/ z ph
Assertion	sb7f	|- ( [ y / x ] ph <-> E. z ( z = y /\ E. x ( x = z /\ ph ) ) )

Proof

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