This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 bitrid ⊢ ( 𝐵 ∈ 𝐶 → ( ∀ 𝑥 ∈ ( 𝐴 ∪ { 𝐵 } ) 𝜑 ↔ ( ∀ 𝑥 ∈ 𝐴 𝜑 ∧ 𝜓 ) ) )