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Theorem ssuniOLD 4396
 Description: Obsolete proof of ssuni 4395 as of 26-Jul-2021. (Contributed by NM, 24-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ssuniOLD ((𝐴𝐵𝐵𝐶) → 𝐴 𝐶)

Proof of Theorem ssuniOLD
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq2 2677 . . . . . . 7 (𝑥 = 𝐵 → (𝑦𝑥𝑦𝐵))
21imbi1d 330 . . . . . 6 (𝑥 = 𝐵 → ((𝑦𝑥𝑦 𝐶) ↔ (𝑦𝐵𝑦 𝐶)))
3 elunii 4377 . . . . . . 7 ((𝑦𝑥𝑥𝐶) → 𝑦 𝐶)
43expcom 450 . . . . . 6 (𝑥𝐶 → (𝑦𝑥𝑦 𝐶))
52, 4vtoclga 3245 . . . . 5 (𝐵𝐶 → (𝑦𝐵𝑦 𝐶))
65imim2d 55 . . . 4 (𝐵𝐶 → ((𝑦𝐴𝑦𝐵) → (𝑦𝐴𝑦 𝐶)))
76alimdv 1832 . . 3 (𝐵𝐶 → (∀𝑦(𝑦𝐴𝑦𝐵) → ∀𝑦(𝑦𝐴𝑦 𝐶)))
8 dfss2 3557 . . 3 (𝐴𝐵 ↔ ∀𝑦(𝑦𝐴𝑦𝐵))
9 dfss2 3557 . . 3 (𝐴 𝐶 ↔ ∀𝑦(𝑦𝐴𝑦 𝐶))
107, 8, 93imtr4g 284 . 2 (𝐵𝐶 → (𝐴𝐵𝐴 𝐶))
1110impcom 445 1 ((𝐴𝐵𝐵𝐶) → 𝐴 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1473   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  ∪ cuni 4372 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547  df-ss 3554  df-uni 4373 This theorem is referenced by: (None)
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