Theorem sbcth2

Description: A substitution into a theorem. (Contributed by NM, 1-Mar-2008) (Proof shortened by Mario Carneiro, 13-Oct-2016)

	Ref	Expression
Hypothesis	sbcth2.1	⊢ ( 𝑥 ∈ 𝐵 → 𝜑 )
Assertion	sbcth2	⊢ ( 𝐴 ∈ 𝐵 → [ 𝐴 / 𝑥 ] 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		sbcth2.1	⊢ ( 𝑥 ∈ 𝐵 → 𝜑 )
2	1	rgen	⊢ ∀ 𝑥 ∈ 𝐵 𝜑
3		rspsbc	⊢ ( 𝐴 ∈ 𝐵 → ( ∀ 𝑥 ∈ 𝐵 𝜑 → [ 𝐴 / 𝑥 ] 𝜑 ) )
4	2 3	mpi	⊢ ( 𝐴 ∈ 𝐵 → [ 𝐴 / 𝑥 ] 𝜑 )