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Theorem ssdifim 4092
 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 4088 . . 3 (𝐴𝑉 ↔ (𝑉 ∖ (𝑉𝐴)) = 𝐴)
2 eqcom 2832 . . 3 ((𝑉 ∖ (𝑉𝐴)) = 𝐴𝐴 = (𝑉 ∖ (𝑉𝐴)))
31, 2sylbb 211 . 2 (𝐴𝑉𝐴 = (𝑉 ∖ (𝑉𝐴)))
4 difeq2 3949 . . 3 (𝐵 = (𝑉𝐴) → (𝑉𝐵) = (𝑉 ∖ (𝑉𝐴)))
54eqcomd 2831 . 2 (𝐵 = (𝑉𝐴) → (𝑉 ∖ (𝑉𝐴)) = (𝑉𝐵))
63, 5sylan9eq 2881 1 ((𝐴𝑉𝐵 = (𝑉𝐴)) → 𝐴 = (𝑉𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   = wceq 1656   ∖ cdif 3795   ⊆ wss 3798 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-ext 2803 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rab 3126  df-v 3416  df-dif 3801  df-in 3805  df-ss 3812 This theorem is referenced by:  ssdifsym  4093  frgrwopregbsn  27687
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