Theorem sbcid

Description: An identity theorem for substitution. See sbid . (Contributed by Mario Carneiro, 18-Feb-2017)

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

Proof

Step	Hyp	Ref	Expression
1		sbsbc	⊢ ( [ 𝑥 / 𝑥 ] 𝜑 ↔ [ 𝑥 / 𝑥 ] 𝜑 )
2		sbid	⊢ ( [ 𝑥 / 𝑥 ] 𝜑 ↔ 𝜑 )
3	1 2	bitr3i	⊢ ( [ 𝑥 / 𝑥 ] 𝜑 ↔ 𝜑 )