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Theorem preqr2g 4126
 Description: Reverse equality lemma for unordered pairs. If two unordered pairs have the same first element, the second elements are equal. (Contributed by SF, 12-Jan-2015.)
Assertion
Ref Expression
preqr2g ((A V B W) → ({C, A} = {C, B} → A = B))

Proof of Theorem preqr2g
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 preq2 3800 . . . 4 (x = A → {C, x} = {C, A})
21eqeq1d 2361 . . 3 (x = A → ({C, x} = {C, y} ↔ {C, A} = {C, y}))
3 eqeq1 2359 . . 3 (x = A → (x = yA = y))
42, 3imbi12d 311 . 2 (x = A → (({C, x} = {C, y} → x = y) ↔ ({C, A} = {C, y} → A = y)))
5 preq2 3800 . . . 4 (y = B → {C, y} = {C, B})
65eqeq2d 2364 . . 3 (y = B → ({C, A} = {C, y} ↔ {C, A} = {C, B}))
7 eqeq2 2362 . . 3 (y = B → (A = yA = B))
86, 7imbi12d 311 . 2 (y = B → (({C, A} = {C, y} → A = y) ↔ ({C, A} = {C, B} → A = B)))
9 vex 2862 . . 3 x V
10 vex 2862 . . 3 y V
119, 10preqr2 4125 . 2 ({C, x} = {C, y} → x = y)
124, 8, 11vtocl2g 2918 1 ((A V B W) → ({C, A} = {C, B} → A = B))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cpr 3738 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214  df-sn 3741  df-pr 3742 This theorem is referenced by:  opkthg  4131
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