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

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 → ( 𝜑 ↔ 𝜓 ) )
Assertion ralsn ( ∀ 𝑥 ∈ { 𝐴 } 𝜑 ↔ 𝜓 )

Proof

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