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Metamath Proof Explorer

Theorem sbie

Description: Conversion of implicit substitution to explicit substitution. For versions requiring disjoint variables, but fewer axioms, see sbiev and sbievw . Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 30-Jun-1994) (Revised by Mario Carneiro, 4-Oct-2016) (Proof shortened by Wolf Lammen, 13-Jul-2019) (New usage is discouraged.)

	Ref	Expression
Hypotheses	sbie.1	⊢ Ⅎ 𝑥 𝜓
	sbie.2	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion	sbie	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		sbie.1	⊢ Ⅎ 𝑥 𝜓
2		sbie.2	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
3		equsb1	⊢ [ 𝑦 / 𝑥 ] 𝑥 = 𝑦
4	2	sbimi	⊢ ( [ 𝑦 / 𝑥 ] 𝑥 = 𝑦 → [ 𝑦 / 𝑥 ] ( 𝜑 ↔ 𝜓 ) )
5	3 4	ax-mp	⊢ [ 𝑦 / 𝑥 ] ( 𝜑 ↔ 𝜓 )
6	1	sbf	⊢ ( [ 𝑦 / 𝑥 ] 𝜓 ↔ 𝜓 )
7	6	sblbis	⊢ ( [ 𝑦 / 𝑥 ] ( 𝜑 ↔ 𝜓 ) ↔ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜓 ) )
8	5 7	mpbi	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜓 )