Theorem rspsbc

Description: Restricted quantifier version of Axiom 4 of Mendelson p. 69. This provides an axiom for a predicate calculus for a restricted domain. This theorem generalizes the unrestricted stdpc4 and spsbc . See also rspsbca and rspcsbela . (Contributed by NM, 17-Nov-2006) (Proof shortened by Mario Carneiro, 13-Oct-2016)

	Ref	Expression
Assertion	rspsbc	⊢ ( 𝐴 ∈ 𝐵 → ( ∀ 𝑥 ∈ 𝐵 𝜑 → [ 𝐴 / 𝑥 ] 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		cbvralsvw	⊢ ( ∀ 𝑥 ∈ 𝐵 𝜑 ↔ ∀ 𝑦 ∈ 𝐵 [ 𝑦 / 𝑥 ] 𝜑 )
2		dfsbcq2	⊢ ( 𝑦 = 𝐴 → ( [ 𝑦 / 𝑥 ] 𝜑 ↔ [ 𝐴 / 𝑥 ] 𝜑 ) )
3	2	rspcv	⊢ ( 𝐴 ∈ 𝐵 → ( ∀ 𝑦 ∈ 𝐵 [ 𝑦 / 𝑥 ] 𝜑 → [ 𝐴 / 𝑥 ] 𝜑 ) )
4	1 3	biimtrid	⊢ ( 𝐴 ∈ 𝐵 → ( ∀ 𝑥 ∈ 𝐵 𝜑 → [ 𝐴 / 𝑥 ] 𝜑 ) )