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Metamath Proof Explorer

Theorem sbiedv

Description: Conversion of implicit substitution to explicit substitution (deduction version of sbie ). Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker sbiedvw when possible. (Contributed by NM, 7-Jan-2017) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	sbiedv.1	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 ↔ 𝜒 ) )
	Assertion	sbiedv	⊢ ( 𝜑 → ( [ 𝑦 / 𝑥 ] 𝜓 ↔ 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		sbiedv.1	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 ↔ 𝜒 ) )
2		nfv	⊢ Ⅎ 𝑥 𝜑
3		nfvd	⊢ ( 𝜑 → Ⅎ 𝑥 𝜒 )
4	1	ex	⊢ ( 𝜑 → ( 𝑥 = 𝑦 → ( 𝜓 ↔ 𝜒 ) ) )
5	2 3 4	sbied	⊢ ( 𝜑 → ( [ 𝑦 / 𝑥 ] 𝜓 ↔ 𝜒 ) )