Theorem ralunsn

Description: Restricted quantification over the union of a set and a singleton, using implicit substitution. (Contributed by Paul Chapman, 17-Nov-2012) (Revised by Mario Carneiro, 23-Apr-2015)

	Ref	Expression
Hypothesis	ralunsn.1	⊢ ( 𝑥 = 𝐵 → ( 𝜑 ↔ 𝜓 ) )
Assertion	ralunsn	⊢ ( 𝐵 ∈ 𝐶 → ( ∀ 𝑥 ∈ ( 𝐴 ∪ { 𝐵 } ) 𝜑 ↔ ( ∀ 𝑥 ∈ 𝐴 𝜑 ∧ 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ralunsn.1	⊢ ( 𝑥 = 𝐵 → ( 𝜑 ↔ 𝜓 ) )
2		ralunb	⊢ ( ∀ 𝑥 ∈ ( 𝐴 ∪ { 𝐵 } ) 𝜑 ↔ ( ∀ 𝑥 ∈ 𝐴 𝜑 ∧ ∀ 𝑥 ∈ { 𝐵 } 𝜑 ) )
3	1	ralsng	⊢ ( 𝐵 ∈ 𝐶 → ( ∀ 𝑥 ∈ { 𝐵 } 𝜑 ↔ 𝜓 ) )
4	3	anbi2d	⊢ ( 𝐵 ∈ 𝐶 → ( ( ∀ 𝑥 ∈ 𝐴 𝜑 ∧ ∀ 𝑥 ∈ { 𝐵 } 𝜑 ) ↔ ( ∀ 𝑥 ∈ 𝐴 𝜑 ∧ 𝜓 ) ) )
5	2 4	syl5bb	⊢ ( 𝐵 ∈ 𝐶 → ( ∀ 𝑥 ∈ ( 𝐴 ∪ { 𝐵 } ) 𝜑 ↔ ( ∀ 𝑥 ∈ 𝐴 𝜑 ∧ 𝜓 ) ) )