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Theorem ssdifim 4272
Description: Implication of a class difference with a subclass. (Contributed by AV, 3-Jan-2022.)
Assertion
Ref Expression
ssdifim ((𝐴𝑉𝐵 = (𝑉𝐴)) → 𝐴 = (𝑉𝐵))

Proof of Theorem ssdifim
StepHypRef Expression
1 dfss4 4268 . . 3 (𝐴𝑉 ↔ (𝑉 ∖ (𝑉𝐴)) = 𝐴)
2 eqcom 2743 . . 3 ((𝑉 ∖ (𝑉𝐴)) = 𝐴𝐴 = (𝑉 ∖ (𝑉𝐴)))
31, 2sylbb 219 . 2 (𝐴𝑉𝐴 = (𝑉 ∖ (𝑉𝐴)))
4 difeq2 4119 . . 3 (𝐵 = (𝑉𝐴) → (𝑉𝐵) = (𝑉 ∖ (𝑉𝐴)))
54eqcomd 2742 . 2 (𝐵 = (𝑉𝐴) → (𝑉 ∖ (𝑉𝐴)) = (𝑉𝐵))
63, 5sylan9eq 2796 1 ((𝐴𝑉𝐵 = (𝑉𝐴)) → 𝐴 = (𝑉𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  cdif 3947  wss 3950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-in 3957  df-ss 3967
This theorem is referenced by:  ssdifsym  4273  frgrwopregbsn  30337
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