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Theorem ssdifim 3895
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 3891 . . 3 (𝐴𝑉 ↔ (𝑉 ∖ (𝑉𝐴)) = 𝐴)
2 eqcom 2658 . . 3 ((𝑉 ∖ (𝑉𝐴)) = 𝐴𝐴 = (𝑉 ∖ (𝑉𝐴)))
31, 2sylbb 209 . 2 (𝐴𝑉𝐴 = (𝑉 ∖ (𝑉𝐴)))
4 difeq2 3755 . . 3 (𝐵 = (𝑉𝐴) → (𝑉𝐵) = (𝑉 ∖ (𝑉𝐴)))
54eqcomd 2657 . 2 (𝐵 = (𝑉𝐴) → (𝑉 ∖ (𝑉𝐴)) = (𝑉𝐵))
63, 5sylan9eq 2705 1 ((𝐴𝑉𝐵 = (𝑉𝐴)) → 𝐴 = (𝑉𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  cdif 3604  wss 3607
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-ral 2946  df-rab 2950  df-v 3233  df-dif 3610  df-in 3614  df-ss 3621
This theorem is referenced by:  ssdifsym  3896  frgrwopregbsn  27297
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