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Metamath Proof Explorer

Theorem sbid2

Description: An identity law for substitution. Usage of this theorem is discouraged because it depends on ax-13 . Check out sbid2vw for a weaker version requiring fewer axioms. (Contributed by NM, 14-May-1993) (Revised by Mario Carneiro, 6-Oct-2016) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	sbid2.1	⊢ Ⅎ 𝑥 𝜑
	Assertion	sbid2	⊢ ( [ 𝑦 / 𝑥 ] [ 𝑥 / 𝑦 ] 𝜑 ↔ 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		sbid2.1	⊢ Ⅎ 𝑥 𝜑
2		sbco	⊢ ( [ 𝑦 / 𝑥 ] [ 𝑥 / 𝑦 ] 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 )
3	1	sbf	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜑 )
4	2 3	bitri	⊢ ( [ 𝑦 / 𝑥 ] [ 𝑥 / 𝑦 ] 𝜑 ↔ 𝜑 )