Theorem sbctt

Description: Substitution for a variable not free in a wff does not affect it. (Contributed by Mario Carneiro, 14-Oct-2016)

	Ref	Expression
Assertion	sbctt	⊢ ( ( 𝐴 ∈ 𝑉 ∧ Ⅎ 𝑥 𝜑 ) → ( [ 𝐴 / 𝑥 ] 𝜑 ↔ 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		dfsbcq2	⊢ ( 𝑦 = 𝐴 → ( [ 𝑦 / 𝑥 ] 𝜑 ↔ [ 𝐴 / 𝑥 ] 𝜑 ) )
2	1	bibi1d	⊢ ( 𝑦 = 𝐴 → ( ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜑 ) ↔ ( [ 𝐴 / 𝑥 ] 𝜑 ↔ 𝜑 ) ) )
3	2	imbi2d	⊢ ( 𝑦 = 𝐴 → ( ( Ⅎ 𝑥 𝜑 → ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜑 ) ) ↔ ( Ⅎ 𝑥 𝜑 → ( [ 𝐴 / 𝑥 ] 𝜑 ↔ 𝜑 ) ) ) )
4		sbft	⊢ ( Ⅎ 𝑥 𝜑 → ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜑 ) )
5	3 4	vtoclg	⊢ ( 𝐴 ∈ 𝑉 → ( Ⅎ 𝑥 𝜑 → ( [ 𝐴 / 𝑥 ] 𝜑 ↔ 𝜑 ) ) )
6	5	imp	⊢ ( ( 𝐴 ∈ 𝑉 ∧ Ⅎ 𝑥 𝜑 ) → ( [ 𝐴 / 𝑥 ] 𝜑 ↔ 𝜑 ) )