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Metamath Proof Explorer

Theorem sbidm

Description: An idempotent law for substitution. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 30-Jun-1994) (Proof shortened by Andrew Salmon, 25-May-2011) (Proof shortened by Wolf Lammen, 13-Jul-2019) (New usage is discouraged.)

		Ref	Expression
	Assertion	sbidm	⊢ ( [ 𝑦 / 𝑥 ] [ 𝑦 / 𝑥 ] 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		sbcom3	⊢ ( [ 𝑦 / 𝑥 ] [ 𝑥 / 𝑥 ] 𝜑 ↔ [ 𝑦 / 𝑥 ] [ 𝑦 / 𝑥 ] 𝜑 )
2		sbid	⊢ ( [ 𝑥 / 𝑥 ] 𝜑 ↔ 𝜑 )
3	2	sbbii	⊢ ( [ 𝑦 / 𝑥 ] [ 𝑥 / 𝑥 ] 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 )
4	1 3	bitr3i	⊢ ( [ 𝑦 / 𝑥 ] [ 𝑦 / 𝑥 ] 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 )