Theorem preqr2 4381

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 4267	. . 3 ⊢ {𝐶, 𝐴} = {𝐴, 𝐶}
2		prcom 4267	. . 3 ⊢ {𝐶, 𝐵} = {𝐵, 𝐶}
3	1, 2	eqeq12i 2636	. 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 4379	. 2 ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵)
7	3, 6	sylbi 207	1 ⊢ ({𝐶, 𝐴} = {𝐶, 𝐵} → 𝐴 = 𝐵)

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  Vcvv 3200  {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:  preq12b  4382  opth  4945  opthreg  8515  usgredgreu  26110  uspgredg2vtxeu  26112  altopthsn  32068