Theorem sb10f

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. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 9-May-2005) (Revised by Mario Carneiro, 6-Oct-2016) (New usage is discouraged.)

	Ref	Expression
Hypothesis	sb10f.1	$$⊢Ⅎx\phi$$
Assertion	sb10f	$$⊢\left[y/z\right]\phi \leftrightarrow \exists x\left(x=y\wedge \left[x/z\right]\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		sb10f.1	$$⊢Ⅎx\phi$$
2	1	nfsb	$$⊢Ⅎx\left[y/z\right]\phi$$
3		sbequ	$$⊢x=y\to \left(\left[x/z\right]\phi \leftrightarrow \left[y/z\right]\phi \right)$$
4	2 3	equsexv	$$⊢\exists x\left(x=y\wedge \left[x/z\right]\phi \right)\leftrightarrow \left[y/z\right]\phi$$
5	4	bicomi	$$⊢\left[y/z\right]\phi \leftrightarrow \exists x\left(x=y\wedge \left[x/z\right]\phi \right)$$