Theorem sb10f 1871

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 1823	. . 3 |- ( [ y / z ] ph -> A. x [ y / z ] ph )
3		sbequ 1721	. . 3 |- ( x = y -> ( [ x / z ] ph <-> [ y / z ] ph ) )
4	2, 3	equsex 1616	. 2 |- ( E. x ( x = y /\ [ x / z ] ph ) <-> [ y / z ] ph )
5	4	bicomi 123	1 |- ( [ y / z ] ph <-> E. x ( x = y /\ [ x / z ] ph ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   A.wal 1241   E.wex 1381   [wsb 1645

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646

This theorem is referenced by: (None)