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

Proof

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