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Theorem ssuniOLD 4612
Description: Obsolete proof of ssuni 4611 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 2828 . . . . . . 7 (𝑥 = 𝐵 → (𝑦𝑥𝑦𝐵))
21imbi1d 330 . . . . . 6 (𝑥 = 𝐵 → ((𝑦𝑥𝑦 𝐶) ↔ (𝑦𝐵𝑦 𝐶)))
3 elunii 4593 . . . . . . 7 ((𝑦𝑥𝑥𝐶) → 𝑦 𝐶)
43expcom 450 . . . . . 6 (𝑥𝐶 → (𝑦𝑥𝑦 𝐶))
52, 4vtoclga 3412 . . . . 5 (𝐵𝐶 → (𝑦𝐵𝑦 𝐶))
65imim2d 57 . . . 4 (𝐵𝐶 → ((𝑦𝐴𝑦𝐵) → (𝑦𝐴𝑦 𝐶)))
76alimdv 1994 . . 3 (𝐵𝐶 → (∀𝑦(𝑦𝐴𝑦𝐵) → ∀𝑦(𝑦𝐴𝑦 𝐶)))
8 dfss2 3732 . . 3 (𝐴𝐵 ↔ ∀𝑦(𝑦𝐴𝑦𝐵))
9 dfss2 3732 . . 3 (𝐴 𝐶 ↔ ∀𝑦(𝑦𝐴𝑦 𝐶))
107, 8, 93imtr4g 285 . 2 (𝐵𝐶 → (𝐴𝐵𝐴 𝐶))
1110impcom 445 1 ((𝐴𝐵𝐵𝐶) → 𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wal 1630   = wceq 1632  wcel 2139  wss 3715   cuni 4588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-in 3722  df-ss 3729  df-uni 4589
This theorem is referenced by: (None)
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