Description: Biconditional equality lemma for unordered pairs, deduction form. Two unordered pairs have the same first element iff the second elements are equal. (Contributed by AV, 18-Dec-2020)
Ref | Expression | ||
---|---|---|---|
Hypotheses | preq1b.a | |- ( ph -> A e. V ) |
|
preq1b.b | |- ( ph -> B e. W ) |
||
Assertion | preq2b | |- ( ph -> ( { C , A } = { C , B } <-> A = B ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preq1b.a | |- ( ph -> A e. V ) |
|
2 | preq1b.b | |- ( ph -> B e. W ) |
|
3 | prcom | |- { C , A } = { A , C } |
|
4 | prcom | |- { C , B } = { B , C } |
|
5 | 3 4 | eqeq12i | |- ( { C , A } = { C , B } <-> { A , C } = { B , C } ) |
6 | 1 2 | preq1b | |- ( ph -> ( { A , C } = { B , C } <-> A = B ) ) |
7 | 5 6 | syl5bb | |- ( ph -> ( { C , A } = { C , B } <-> A = B ) ) |