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Theorem sneqrg 4839
Description: Closed form of sneqr 4840. (Contributed by Scott Fenton, 1-Apr-2011.) (Proof shortened by JJ, 23-Jul-2021.)
Assertion
Ref Expression
sneqrg (𝐴𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵))

Proof of Theorem sneqrg
StepHypRef Expression
1 snidg 4660 . . 3 (𝐴𝑉𝐴 ∈ {𝐴})
2 eleq2 2830 . . 3 ({𝐴} = {𝐵} → (𝐴 ∈ {𝐴} ↔ 𝐴 ∈ {𝐵}))
31, 2syl5ibcom 245 . 2 (𝐴𝑉 → ({𝐴} = {𝐵} → 𝐴 ∈ {𝐵}))
4 elsng 4640 . 2 (𝐴𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
53, 4sylibd 239 1 (𝐴𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  {csn 4626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-sn 4627
This theorem is referenced by:  sneqr  4840  sneqbg  4843  snsssng  32533  preimane  32680  altopth1  35966  altopth2  35967
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