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Metamath Proof Explorer

Theorem sbcgf

Description: Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 11-Oct-2004) (Proof shortened by Andrew Salmon, 29-Jun-2011)

		Ref	Expression
	Hypothesis	sbcgf.1	⊢ Ⅎ 𝑥 𝜑
	Assertion	sbcgf	⊢ ( 𝐴 ∈ 𝑉 → ( [ 𝐴 / 𝑥 ] 𝜑 ↔ 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		sbcgf.1	⊢ Ⅎ 𝑥 𝜑
2		sbctt	⊢ ( ( 𝐴 ∈ 𝑉 ∧ Ⅎ 𝑥 𝜑 ) → ( [ 𝐴 / 𝑥 ] 𝜑 ↔ 𝜑 ) )
3	1 2	mpan2	⊢ ( 𝐴 ∈ 𝑉 → ( [ 𝐴 / 𝑥 ] 𝜑 ↔ 𝜑 ) )