Theorem preqr2 4818

Description: Reverse equality lemma for unordered pairs. If two unordered pairs have the same first element, the second elements are equal. (Contributed by NM, 15-Jul-1993.)

Hypotheses

Ref	Expression
preqr1.a	⊢ 𝐴 ∈ V
preqr1.b	⊢ 𝐵 ∈ V

Assertion

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

Proof of Theorem preqr2

Step	Hyp	Ref	Expression
1		prcom 4703	. . 3 ⊢ {𝐶, 𝐴} = {𝐴, 𝐶}
2		prcom 4703	. . 3 ⊢ {𝐶, 𝐵} = {𝐵, 𝐶}
3	1, 2	eqeq12i 2787	. 2 ⊢ ({𝐶, 𝐴} = {𝐶, 𝐵} ↔ {𝐴, 𝐶} = {𝐵, 𝐶})
4		preqr1.a	. . 3 ⊢ 𝐴 ∈ V
5		preqr1.b	. . 3 ⊢ 𝐵 ∈ V
6	4, 5	preqr1 4817	. 2 ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵)
7	3, 6	sylbi 220	1 ⊢ ({𝐶, 𝐴} = {𝐶, 𝐵} → 𝐴 = 𝐵)

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  Vcvv 3463  {cpr 4596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-un 3918  df-sn 4595  df-pr 4597

This theorem is referenced by:  preq12b  4819  opth  5459  opthreg  9586  usgredgreu  29508  uspgredg2vtxeu  29510  altopthsn  36351