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Metamath Proof Explorer

Theorem preq1b

Description: Biconditional equality lemma for unordered pairs, deduction form. Two unordered pairs have the same second element iff the first 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 preq1b 
|- ( ph -> ( { A , C } = { B , C } <-> A = B ) )

Proof

Step Hyp Ref Expression
1 preq1b.a 
 |-  ( ph -> A e. V )
2 preq1b.b 
 |-  ( ph -> B e. W )
3 prid1g 
 |-  ( A e. V -> A e. { A , C } )
4 1 3 syl 
 |-  ( ph -> A e. { A , C } )
5 eleq2 
 |-  ( { A , C } = { B , C } -> ( A e. { A , C } <-> A e. { B , C } ) )
6 4 5 syl5ibcom 
 |-  ( ph -> ( { A , C } = { B , C } -> A e. { B , C } ) )
7 elprg 
 |-  ( A e. V -> ( A e. { B , C } <-> ( A = B \/ A = C ) ) )
8 1 7 syl 
 |-  ( ph -> ( A e. { B , C } <-> ( A = B \/ A = C ) ) )
9 6 8 sylibd 
 |-  ( ph -> ( { A , C } = { B , C } -> ( A = B \/ A = C ) ) )
10 9 imp 
 |-  ( ( ph /\ { A , C } = { B , C } ) -> ( A = B \/ A = C ) )
11 prid1g 
 |-  ( B e. W -> B e. { B , C } )
12 2 11 syl 
 |-  ( ph -> B e. { B , C } )
13 eleq2 
 |-  ( { A , C } = { B , C } -> ( B e. { A , C } <-> B e. { B , C } ) )
14 12 13 syl5ibrcom 
 |-  ( ph -> ( { A , C } = { B , C } -> B e. { A , C } ) )
15 elprg 
 |-  ( B e. W -> ( B e. { A , C } <-> ( B = A \/ B = C ) ) )
16 2 15 syl 
 |-  ( ph -> ( B e. { A , C } <-> ( B = A \/ B = C ) ) )
17 14 16 sylibd 
 |-  ( ph -> ( { A , C } = { B , C } -> ( B = A \/ B = C ) ) )
18 17 imp 
 |-  ( ( ph /\ { A , C } = { B , C } ) -> ( B = A \/ B = C ) )
19 eqcom 
 |-  ( A = B <-> B = A )
20 eqeq2 
 |-  ( A = C -> ( B = A <-> B = C ) )
21 10 18 19 20 oplem1 
 |-  ( ( ph /\ { A , C } = { B , C } ) -> A = B )
22 21 ex 
 |-  ( ph -> ( { A , C } = { B , C } -> A = B ) )
23 preq1 
 |-  ( A = B -> { A , C } = { B , C } )
24 22 23 impbid1 
 |-  ( ph -> ( { A , C } = { B , C } <-> A = B ) )