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Theorem sneqrgOLD 4510
Description: Obsolete proof of sneqrg 4507 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 4323 . . . 4 (𝑥 = 𝐴 → {𝑥} = {𝐴})
21eqeq1d 2754 . . 3 (𝑥 = 𝐴 → ({𝑥} = {𝐵} ↔ {𝐴} = {𝐵}))
3 eqeq1 2756 . . 3 (𝑥 = 𝐴 → (𝑥 = 𝐵𝐴 = 𝐵))
42, 3imbi12d 333 . 2 (𝑥 = 𝐴 → (({𝑥} = {𝐵} → 𝑥 = 𝐵) ↔ ({𝐴} = {𝐵} → 𝐴 = 𝐵)))
5 vex 3335 . . 3 𝑥 ∈ V
65sneqr 4508 . 2 ({𝑥} = {𝐵} → 𝑥 = 𝐵)
74, 6vtoclg 3398 1 (𝐴𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1624  wcel 2131  {csn 4313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-v 3334  df-sn 4314
This theorem is referenced by: (None)
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