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Metamath Proof Explorer

Theorem brcnv

Description: The converse of a binary relation swaps arguments. Theorem 11 of Suppes p. 61. (Contributed by NM, 13-Aug-1995)

Ref Expression
Hypotheses opelcnv.1 ⊢ 𝐴 ∈ V
opelcnv.2 ⊢ 𝐵 ∈ V
Assertion brcnv ( 𝐴 ◡ 𝑅 𝐵 ↔ 𝐵 𝑅 𝐴 )

Proof

Step Hyp Ref Expression
1 opelcnv.1 ⊢ 𝐴 ∈ V
2 opelcnv.2 ⊢ 𝐵 ∈ V
3 brcnvg ⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( 𝐴 ◡ 𝑅 𝐵 ↔ 𝐵 𝑅 𝐴 ) )
4 1 2 3 mp2an ⊢ ( 𝐴 ◡ 𝑅 𝐵 ↔ 𝐵 𝑅 𝐴 )