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Metamath Proof Explorer

Theorem sbcgfi

Description: Substitution for a variable not free in a wff does not affect it, in inference form. (Contributed by Giovanni Mascellani, 1-Jun-2019)

	Ref	Expression
Hypotheses	sbcgfi.1	⊢ 𝐴 ∈ V
	sbcgfi.2	⊢ Ⅎ 𝑥 𝜑
Assertion	sbcgfi	⊢ ( [ 𝐴 / 𝑥 ] 𝜑 ↔ 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		sbcgfi.1	⊢ 𝐴 ∈ V
2		sbcgfi.2	⊢ Ⅎ 𝑥 𝜑
3	2	sbcgf	⊢ ( 𝐴 ∈ V → ( [ 𝐴 / 𝑥 ] 𝜑 ↔ 𝜑 ) )
4	1 3	ax-mp	⊢ ( [ 𝐴 / 𝑥 ] 𝜑 ↔ 𝜑 )