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Metamath Proof Explorer

Theorem 3brtr4d

Description: Substitution of equality into both sides of a binary relation. (Contributed by NM, 21-Feb-2005)

Ref Expression
Hypotheses 3brtr4d.1 ⊢ ( 𝜑 → 𝐴 𝑅 𝐵 )
3brtr4d.2 ⊢ ( 𝜑 → 𝐶 = 𝐴 )
3brtr4d.3 ⊢ ( 𝜑 → 𝐷 = 𝐵 )
Assertion 3brtr4d ( 𝜑 → 𝐶 𝑅 𝐷 )

Proof

Step Hyp Ref Expression
1 3brtr4d.1 ⊢ ( 𝜑 → 𝐴 𝑅 𝐵 )
2 3brtr4d.2 ⊢ ( 𝜑 → 𝐶 = 𝐴 )
3 3brtr4d.3 ⊢ ( 𝜑 → 𝐷 = 𝐵 )
4 2 3 breq12d ⊢ ( 𝜑 → ( 𝐶 𝑅 𝐷 ↔ 𝐴 𝑅 𝐵 ) )
5 1 4 mpbird ⊢ ( 𝜑 → 𝐶 𝑅 𝐷 )