Metamath Proof Explorer


Theorem sb7h

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) (Proof shortened by Andrew Salmon, 25-May-2011) (New usage is discouraged.)

Ref Expression
Hypothesis sb7h.1 φ z φ
Assertion sb7h y x φ z z = y x x = z φ

Proof

Step Hyp Ref Expression
1 sb7h.1 φ z φ
2 1 nf5i z φ
3 2 sb7f y x φ z z = y x x = z φ