This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 ) )