Theorem sb10f 2014

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 1968	. . 3 |- ( [ y / z ] ph -> A. x [ y / z ] ph )
3		sbequ 1854	. . 3 |- ( x = y -> ( [ x / z ] ph <-> [ y / z ] ph ) )
4	2, 3	equsex 1742	. 2 |- ( E. x ( x = y /\ [ x / z ] ph ) <-> [ y / z ] ph )
5	4	bicomi 132	1 |- ( [ y / z ] ph <-> E. x ( x = y /\ [ x / z ] ph ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1362   E.wex 1506   [wsb 1776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777

This theorem is referenced by: (None)