Metamath Proof Explorer


Theorem ralsn

Description: Convert a universal quantification restricted to a singleton to a substitution. (Contributed by NM, 27-Apr-2009)

Ref Expression
Hypotheses ralsn.1 𝐴 ∈ V
ralsn.2 ( 𝑥 = 𝐴 → ( 𝜑𝜓 ) )
Assertion ralsn ( ∀ 𝑥 ∈ { 𝐴 } 𝜑𝜓 )

Proof

Step Hyp Ref Expression
1 ralsn.1 𝐴 ∈ V
2 ralsn.2 ( 𝑥 = 𝐴 → ( 𝜑𝜓 ) )
3 2 ralsng ( 𝐴 ∈ V → ( ∀ 𝑥 ∈ { 𝐴 } 𝜑𝜓 ) )
4 1 3 ax-mp ( ∀ 𝑥 ∈ { 𝐴 } 𝜑𝜓 )