Theorem preq1b 4377

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.)

Hypotheses

Ref	Expression
preq1b.a	|- ( ph -> A e. V )
preq1b.b	|- ( ph -> B e. W )

Assertion

Ref	Expression
preq1b	|- ( ph -> ( { A , C } = { B , C } <-> A = B ) )

Proof of Theorem preq1b

Step	Hyp	Ref	Expression
1		preq1b.a	. . . . . . . 8 |- ( ph -> A e. V )
2		prid1g 4295	. . . . . . . 8 |- ( A e. V -> A e. { A , C } )
3	1, 2	syl 17	. . . . . . 7 |- ( ph -> A e. { A , C } )
4		eleq2 2690	. . . . . . 7 |- ( { A , C } = { B , C } -> ( A e. { A , C } <-> A e. { B , C } ) )
5	3, 4	syl5ibcom 235	. . . . . 6 |- ( ph -> ( { A , C } = { B , C } -> A e. { B , C } ) )
6		elprg 4196	. . . . . . 7 |- ( A e. V -> ( A e. { B , C } <-> ( A = B \/ A = C ) ) )
7	1, 6	syl 17	. . . . . 6 |- ( ph -> ( A e. { B , C } <-> ( A = B \/ A = C ) ) )
8	5, 7	sylibd 229	. . . . 5 |- ( ph -> ( { A , C } = { B , C } -> ( A = B \/ A = C ) ) )
9	8	imp 445	. . . 4 |- ( ( ph /\ { A , C } = { B , C } ) -> ( A = B \/ A = C ) )
10		preq1b.b	. . . . . . . 8 |- ( ph -> B e. W )
11		prid1g 4295	. . . . . . . 8 |- ( B e. W -> B e. { B , C } )
12	10, 11	syl 17	. . . . . . 7 |- ( ph -> B e. { B , C } )
13		eleq2 2690	. . . . . . 7 |- ( { A , C } = { B , C } -> ( B e. { A , C } <-> B e. { B , C } ) )
14	12, 13	syl5ibrcom 237	. . . . . 6 |- ( ph -> ( { A , C } = { B , C } -> B e. { A , C } ) )
15		elprg 4196	. . . . . . 7 |- ( B e. W -> ( B e. { A , C } <-> ( B = A \/ B = C ) ) )
16	10, 15	syl 17	. . . . . 6 |- ( ph -> ( B e. { A , C } <-> ( B = A \/ B = C ) ) )
17	14, 16	sylibd 229	. . . . 5 |- ( ph -> ( { A , C } = { B , C } -> ( B = A \/ B = C ) ) )
18	17	imp 445	. . . 4 |- ( ( ph /\ { A , C } = { B , C } ) -> ( B = A \/ B = C ) )
19		eqcom 2629	. . . 4 |- ( A = B <-> B = A )
20		eqeq2 2633	. . . 4 |- ( A = C -> ( B = A <-> B = C ) )
21	9, 18, 19, 20	oplem1 1007	. . 3 |- ( ( ph /\ { A , C } = { B , C } ) -> A = B )
22	21	ex 450	. 2 |- ( ph -> ( { A , C } = { B , C } -> A = B ) )
23		preq1 4268	. 2 |- ( A = B -> { A , C } = { B , C } )
24	22, 23	impbid1 215	1 |- ( ph -> ( { A , C } = { B , C } <-> A = B ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   {cpr 4179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579  df-sn 4178  df-pr 4180

This theorem is referenced by:  preq2b  4378  preqr1  4379  preqr1g  4385  uhgr3cyclexlem  27041