Theorem sb10f 2456

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. (Contributed by NM, 9-May-2005.) (Revised by Mario Carneiro, 6-Oct-2016.)

Hypothesis

Ref	Expression
sb10f.1	|- F/ x ph

Assertion

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

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y, z)

Proof of Theorem sb10f

Step	Hyp	Ref	Expression
1		sb10f.1	. . . 4 |- F/ x ph
2	1	nfsb 2440	. . 3 |- F/ x [ y / z ] ph
3		sbequ 2376	. . 3 |- ( x = y -> ( [ x / z ] ph <-> [ y / z ] ph ) )
4	2, 3	equsexv 2109	. 2 |- ( E. x ( x = y /\ [ x / z ] ph ) <-> [ y / z ] ph )
5	4	bicomi 214	1 |- ( [ y / z ] ph <-> E. x ( x = y /\ [ x / z ] ph ) )

Syntax hints:   <-> wb 196   /\ wa 384   E.wex 1704   F/wnf 1708   [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by: (None)