Theorem sb10f

Description: Hao Wang's identity axiom P6 in Irving Copi,Symbolic Logic (5th ed., 1979), p. 328. In traditional predicate calculus, this is a sole axiom for identity from which the usual ones can be derived. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 9-May-2005) (Revised by Mario Carneiro, 6-Oct-2016) (New usage is discouraged.)

	Ref	Expression
Hypothesis	sb10f.1	|- F/ x ph
Assertion	sb10f	|- ( [ y / z ] ph <-> E. x ( x = y /\ [ x / z ] ph ) )

Proof

Step	Hyp	Ref	Expression
1		sb10f.1	|- F/ x ph
2	1	nfsb	|- F/ x [ y / z ] ph
3		sbequ	|- ( x = y -> ( [ x / z ] ph <-> [ y / z ] ph ) )
4	2 3	equsexv	|- ( E. x ( x = y /\ [ x / z ] ph ) <-> [ y / z ] ph )
5	4	bicomi	|- ( [ y / z ] ph <-> E. x ( x = y /\ [ x / z ] ph ) )