Database
ZF (ZERMELO-FRAENKEL) SET THEORY
ZF Set Theory - start with the Axiom of Extensionality
Binary relations
breq12
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breqi
Metamath Proof Explorer
Ascii
Unicode
Theorem
breq12
Description:
Equality theorem for a binary relation.
(Contributed by
NM
, 8-Feb-1996)
Ref
Expression
Assertion
breq12
⊢
A
=
B
∧
C
=
D
→
A
R
C
↔
B
R
D
Proof
Step
Hyp
Ref
Expression
1
breq1
⊢
A
=
B
→
A
R
C
↔
B
R
C
2
breq2
⊢
C
=
D
→
B
R
C
↔
B
R
D
3
1
2
sylan9bb
⊢
A
=
B
∧
C
=
D
→
A
R
C
↔
B
R
D