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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