Theorem rspcdf

Description: Restricted specialization, using implicit substitution. (Contributed by Emmett Weisz, 16-Jan-2020)

	Ref	Expression
Hypotheses	rspcdf.1	⊢ Ⅎ 𝑥 𝜑
	rspcdf.2	⊢ Ⅎ 𝑥 𝜒
	rspcdf.3	⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
	rspcdf.4	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
Assertion	rspcdf	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐵 𝜓 → 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		rspcdf.1	⊢ Ⅎ 𝑥 𝜑
2		rspcdf.2	⊢ Ⅎ 𝑥 𝜒
3		rspcdf.3	⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
4		rspcdf.4	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
5	4	ex	⊢ ( 𝜑 → ( 𝑥 = 𝐴 → ( 𝜓 ↔ 𝜒 ) ) )
6	1 5	alrimi	⊢ ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜓 ↔ 𝜒 ) ) )
7	2	rspct	⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜓 ↔ 𝜒 ) ) → ( 𝐴 ∈ 𝐵 → ( ∀ 𝑥 ∈ 𝐵 𝜓 → 𝜒 ) ) )
8	6 3 7	sylc	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐵 𝜓 → 𝜒 ) )