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Metamath Proof Explorer

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	$⊢ Ⅎ x φ$
	Assertion	sb10f	$⊢ [y/ z] φ \leftrightarrow \exists x (x = y \land [x/ z] φ)$

Proof

Step	Hyp	Ref	Expression
1		sb10f.1	$⊢ Ⅎ x φ$
2	1	nfsb	$⊢ Ⅎ x [y/ z] φ$
3		sbequ	$⊢ x = y \to ([x/ z] φ \leftrightarrow [y/ z] φ)$
4	2 3	equsexv	$⊢ \exists x (x = y \land [x/ z] φ) \leftrightarrow [y/ z] φ$
5	4	bicomi	$⊢ [y/ z] φ \leftrightarrow \exists x (x = y \land [x/ z] φ)$