Theorem sb10f 1342

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.

Hypothesis

Ref	Expression
sb10f.1	|- (ph -> A.xph)

Assertion

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

Distinct variable group:   x,y

Proof of Theorem sb10f

Step	Hyp	Ref	Expression
1		sb10f.1	. . . 4 |- (ph -> A.xph)
2	1	hbsb 1333	. . 3 |- ([y / z]ph -> A.x[y / z]ph)
3		sbequ 1229	. . 3 |- (x = y -> ([x / z]ph <-> [y / z]ph))
4	2, 3	equsex 1152	. 2 |- (E.x(x = y /\ [x / z]ph) <-> [y / z]ph)
5	4	bicomi 172	1 |- ([y / z]ph <-> E.x(x = y /\ [x / z]ph))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 954   = wceq 956  E.wex 980  [wsbc 1170

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-12 968  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172

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