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Metamath Proof Explorer

Theorem sbf

Description: Substitution for a variable not free in a wff does not affect it. For a version requiring disjoint variables but fewer axioms, see sbv . (Contributed by NM, 14-May-1993) (Revised by Mario Carneiro, 4-Oct-2016)

		Ref	Expression
	Hypothesis	sbf.1	⊢ Ⅎ 𝑥 𝜑
	Assertion	sbf	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		sbf.1	⊢ Ⅎ 𝑥 𝜑
2		sbft	⊢ ( Ⅎ 𝑥 𝜑 → ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜑 ) )
3	1 2	ax-mp	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜑 )