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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 ↔ ∃ 𝑥 ( 𝑥 = 𝑦 ∧ [ 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 → ( [ 𝑥 / 𝑧 ] 𝜑 ↔ [ 𝑦 / 𝑧 ] 𝜑 ) )
4	2 3	equsexv	⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ [ 𝑥 / 𝑧 ] 𝜑 ) ↔ [ 𝑦 / 𝑧 ] 𝜑 )
5	4	bicomi	⊢ ( [ 𝑦 / 𝑧 ] 𝜑 ↔ ∃ 𝑥 ( 𝑥 = 𝑦 ∧ [ 𝑥 / 𝑧 ] 𝜑 ) )