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Theorem sneqrg 3636
 Description: Closed form of sneqr 3634. (Contributed by Scott Fenton, 1-Apr-2011.)
Assertion
Ref Expression
sneqrg (𝐴𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵))

Proof of Theorem sneqrg
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 sneq 3485 . . . 4 (𝑥 = 𝐴 → {𝑥} = {𝐴})
21eqeq1d 2108 . . 3 (𝑥 = 𝐴 → ({𝑥} = {𝐵} ↔ {𝐴} = {𝐵}))
3 eqeq1 2106 . . 3 (𝑥 = 𝐴 → (𝑥 = 𝐵𝐴 = 𝐵))
42, 3imbi12d 233 . 2 (𝑥 = 𝐴 → (({𝑥} = {𝐵} → 𝑥 = 𝐵) ↔ ({𝐴} = {𝐵} → 𝐴 = 𝐵)))
5 vex 2644 . . 3 𝑥 ∈ V
65sneqr 3634 . 2 ({𝑥} = {𝐵} → 𝑥 = 𝐵)
74, 6vtoclg 2701 1 (𝐴𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1299   ∈ wcel 1448  {csn 3474 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082 This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-sn 3480 This theorem is referenced by:  sneqbg  3637
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