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Theorem sneqrgOLD 4405
 Description: Obsolete proof of sneqrg 4402 as of 23-Jul-2021. (Contributed by Scott Fenton, 1-Apr-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
sneqrgOLD (𝐴𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵))

Proof of Theorem sneqrgOLD
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 sneq 4220 . . . 4 (𝑥 = 𝐴 → {𝑥} = {𝐴})
21eqeq1d 2653 . . 3 (𝑥 = 𝐴 → ({𝑥} = {𝐵} ↔ {𝐴} = {𝐵}))
3 eqeq1 2655 . . 3 (𝑥 = 𝐴 → (𝑥 = 𝐵𝐴 = 𝐵))
42, 3imbi12d 333 . 2 (𝑥 = 𝐴 → (({𝑥} = {𝐵} → 𝑥 = 𝐵) ↔ ({𝐴} = {𝐵} → 𝐴 = 𝐵)))
5 vex 3234 . . 3 𝑥 ∈ V
65sneqr 4403 . 2 ({𝑥} = {𝐵} → 𝑥 = 𝐵)
74, 6vtoclg 3297 1 (𝐴𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1523   ∈ wcel 2030  {csn 4210 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-sn 4211 This theorem is referenced by: (None)
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