Theorem preqr2 3799

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 3698	. . 3 ⊢ {𝐶, 𝐴} = {𝐴, 𝐶}
2		prcom 3698	. . 3 ⊢ {𝐶, 𝐵} = {𝐵, 𝐶}
3	1, 2	eqeq12i 2210	. 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 3798	. 2 ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵)
7	3, 6	sylbi 121	1 ⊢ ({𝐶, 𝐴} = {𝐶, 𝐵} → 𝐴 = 𝐵)

Syntax hints:   → wi 4   = wceq 1364   ∈ wcel 2167  Vcvv 2763  {cpr 3623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-sn 3628  df-pr 3629

This theorem is referenced by:  preq12b  3800  opth  4270  opthreg  4592