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Theorem sneqrg 4750
Description: Closed form of sneqr 4751. (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 4575 . . 3 (𝐴𝑉𝐴 ∈ {𝐴})
2 eleq2 2826 . . 3 ({𝐴} = {𝐵} → (𝐴 ∈ {𝐴} ↔ 𝐴 ∈ {𝐵}))
31, 2syl5ibcom 248 . 2 (𝐴𝑉 → ({𝐴} = {𝐵} → 𝐴 ∈ {𝐵}))
4 elsng 4555 . 2 (𝐴𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
53, 4sylibd 242 1 (𝐴𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1543  wcel 2110  {csn 4541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-sn 4542
This theorem is referenced by:  sneqr  4751  sneqbg  4754  snsssng  30579  preimane  30727  altopth1  34004  altopth2  34005
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