Theorem bj-sbidmOLD

Description: Obsolete proof of sbidm temporarily kept here to check it gives no additional insight. (Contributed by NM, 8-Mar-1995) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	bj-sbidmOLD	⊢ ( [ 𝑦 / 𝑥 ] [ 𝑦 / 𝑥 ] 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		equsb2	⊢ [ 𝑦 / 𝑥 ] 𝑦 = 𝑥
2		sbequ12r	⊢ ( 𝑦 = 𝑥 → ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜑 ) )
3	2	sbimi	⊢ ( [ 𝑦 / 𝑥 ] 𝑦 = 𝑥 → [ 𝑦 / 𝑥 ] ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜑 ) )
4	1 3	ax-mp	⊢ [ 𝑦 / 𝑥 ] ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜑 )
5		sbbi	⊢ ( [ 𝑦 / 𝑥 ] ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜑 ) ↔ ( [ 𝑦 / 𝑥 ] [ 𝑦 / 𝑥 ] 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 ) )
6	4 5	mpbi	⊢ ( [ 𝑦 / 𝑥 ] [ 𝑦 / 𝑥 ] 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 )