Theorem preqr2 4777

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 4665	. . 3 ⊢ {𝐶, 𝐴} = {𝐴, 𝐶}
2		prcom 4665	. . 3 ⊢ {𝐶, 𝐵} = {𝐵, 𝐶}
3	1, 2	eqeq12i 2756	. 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 4776	. 2 ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵)
7	3, 6	sylbi 216	1 ⊢ ({𝐶, 𝐴} = {𝐶, 𝐵} → 𝐴 = 𝐵)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  Vcvv 3422  {cpr 4560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-un 3888  df-sn 4559  df-pr 4561

This theorem is referenced by:  preq12b  4778  opth  5385  opthreg  9306  usgredgreu  27488  uspgredg2vtxeu  27490  altopthsn  34190