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Metamath Proof Explorer

Theorem 3brtr3d

Description: Substitution of equality into both sides of a binary relation. (Contributed by NM, 18-Oct-1999)

Ref Expression
Hypotheses 3brtr3d.1 ⊢ ( 𝜑 → 𝐴 𝑅 𝐵 )
3brtr3d.2 ⊢ ( 𝜑 → 𝐴 = 𝐶 )
3brtr3d.3 ⊢ ( 𝜑 → 𝐵 = 𝐷 )
Assertion 3brtr3d ( 𝜑 → 𝐶 𝑅 𝐷 )

Proof

Step Hyp Ref Expression
1 3brtr3d.1 ⊢ ( 𝜑 → 𝐴 𝑅 𝐵 )
2 3brtr3d.2 ⊢ ( 𝜑 → 𝐴 = 𝐶 )
3 3brtr3d.3 ⊢ ( 𝜑 → 𝐵 = 𝐷 )
4 2 3 breq12d ⊢ ( 𝜑 → ( 𝐴 𝑅 𝐵 ↔ 𝐶 𝑅 𝐷 ) )
5 1 4 mpbid ⊢ ( 𝜑 → 𝐶 𝑅 𝐷 )