Theorem preqr2 3704

Description: Reverse equality lemma for unordered pairs. If two unordered pairs have the same first element, the second elements are equal. (Contributed by NM, 5-Aug-1993.)

Hypotheses

Ref	Expression
preqr2.1	⊢ 𝐴 ∈ V
preqr2.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
preqr2	⊢ ({𝐶, 𝐴} = {𝐶, 𝐵} → 𝐴 = 𝐵)

Proof of Theorem preqr2

Step	Hyp	Ref	Expression
1		prcom 3607	. . 3 ⊢ {𝐶, 𝐴} = {𝐴, 𝐶}
2		prcom 3607	. . 3 ⊢ {𝐶, 𝐵} = {𝐵, 𝐶}
3	1, 2	eqeq12i 2154	. 2 ⊢ ({𝐶, 𝐴} = {𝐶, 𝐵} ↔ {𝐴, 𝐶} = {𝐵, 𝐶})
4		preqr2.1	. . 3 ⊢ 𝐴 ∈ V
5		preqr2.2	. . 3 ⊢ 𝐵 ∈ V
6	4, 5	preqr1 3703	. 2 ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵)
7	3, 6	sylbi 120	1 ⊢ ({𝐶, 𝐴} = {𝐶, 𝐵} → 𝐴 = 𝐵)

Syntax hints:   → wi 4   = wceq 1332   ∈ wcel 1481  Vcvv 2689  {cpr 3533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-sn 3538  df-pr 3539

This theorem is referenced by:  preq12b  3705  opth  4167  opthreg  4479