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Metamath Proof Explorer

Theorem 3brtr3g

Description: Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997)

Ref Expression
Hypotheses 3brtr3g.1 ⊢ ( 𝜑 → 𝐴 𝑅 𝐵 )
3brtr3g.2 ⊢ 𝐴 = 𝐶
3brtr3g.3 ⊢ 𝐵 = 𝐷
Assertion 3brtr3g ( 𝜑 → 𝐶 𝑅 𝐷 )

Proof

Step Hyp Ref Expression
1 3brtr3g.1 ⊢ ( 𝜑 → 𝐴 𝑅 𝐵 )
2 3brtr3g.2 ⊢ 𝐴 = 𝐶
3 3brtr3g.3 ⊢ 𝐵 = 𝐷
4 2 3 breq12i ⊢ ( 𝐴 𝑅 𝐵 ↔ 𝐶 𝑅 𝐷 )
5 1 4 sylib ⊢ ( 𝜑 → 𝐶 𝑅 𝐷 )