Theorem sb10f 1988

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.)

Hypothesis

Ref	Expression
sb10f.1	|- ( ph -> A. 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 |- ( ph -> A. x ph )
2	1	hbsb 1942	. . 3 |- ( [ y / z ] ph -> A. x [ y / z ] ph )
3		sbequ 1833	. . 3 |- ( x = y -> ( [ x / z ] ph <-> [ y / z ] ph ) )
4	2, 3	equsex 1721	. 2 |- ( E. x ( x = y /\ [ x / z ] ph ) <-> [ y / z ] ph )
5	4	bicomi 131	1 |- ( [ y / z ] ph <-> E. x ( x = y /\ [ x / z ] ph ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1346   E.wex 1485   [wsb 1755

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756

This theorem is referenced by: (None)