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

Metamath Proof Explorer

Theorem braba

Description: The law of concretion for a binary relation. (Contributed by NM, 19-Dec-2013)

Ref Expression
Hypotheses opelopaba.1 ⊢ 𝐴 ∈ V
opelopaba.2 ⊢ 𝐵 ∈ V
opelopaba.3 ⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜑 ↔ 𝜓 ) )
braba.4 ⊢ 𝑅 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 }
Assertion braba ( 𝐴 𝑅 𝐵 ↔ 𝜓 )

Proof

Step Hyp Ref Expression
1 opelopaba.1 ⊢ 𝐴 ∈ V
2 opelopaba.2 ⊢ 𝐵 ∈ V
3 opelopaba.3 ⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜑 ↔ 𝜓 ) )
4 braba.4 ⊢ 𝑅 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 }
5 3 4 brabga ⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( 𝐴 𝑅 𝐵 ↔ 𝜓 ) )
6 1 2 5 mp2an ⊢ ( 𝐴 𝑅 𝐵 ↔ 𝜓 )