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

Metamath Proof Explorer

Theorem rexsn

Description: Convert an existential quantification restricted to a singleton to a substitution. (Contributed by Jeff Madsen, 5-Jan-2011)

Ref Expression
Hypotheses ralsn.1 ⊢ 𝐴 ∈ V
ralsn.2 ⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
Assertion rexsn ( ∃ 𝑥 ∈ { 𝐴 } 𝜑 ↔ 𝜓 )

Proof

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