Metamath Proof Explorer


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 ( ( 𝐴𝑉 ∧ Ⅎ 𝑥 𝜑 ) → ( [ 𝐴 / 𝑥 ] 𝜑𝜑 ) )

Proof

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