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Theorem cotrgOLD 6066
Description: Obsolete version of cotrg 6065 as of 29-Dec-2024. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Generalized from its special instance cotr 6068. (Revised by Richard Penner, 24-Dec-2019.) (Proof shortened by SN, 19-Dec-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
cotrgOLD ((𝐴𝐵) ⊆ 𝐶 ↔ ∀𝑥𝑦𝑧((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧

Proof of Theorem cotrgOLD
StepHypRef Expression
1 relco 6064 . . 3 Rel (𝐴𝐵)
2 ssrel3 5746 . . 3 (Rel (𝐴𝐵) → ((𝐴𝐵) ⊆ 𝐶 ↔ ∀𝑥𝑧(𝑥(𝐴𝐵)𝑧𝑥𝐶𝑧)))
31, 2ax-mp 5 . 2 ((𝐴𝐵) ⊆ 𝐶 ↔ ∀𝑥𝑧(𝑥(𝐴𝐵)𝑧𝑥𝐶𝑧))
4 vex 3451 . . . . . . . 8 𝑥 ∈ V
5 vex 3451 . . . . . . . 8 𝑧 ∈ V
64, 5brco 5830 . . . . . . 7 (𝑥(𝐴𝐵)𝑧 ↔ ∃𝑦(𝑥𝐵𝑦𝑦𝐴𝑧))
76imbi1i 350 . . . . . 6 ((𝑥(𝐴𝐵)𝑧𝑥𝐶𝑧) ↔ (∃𝑦(𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
8 19.23v 1946 . . . . . 6 (∀𝑦((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ (∃𝑦(𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
97, 8bitr4i 278 . . . . 5 ((𝑥(𝐴𝐵)𝑧𝑥𝐶𝑧) ↔ ∀𝑦((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
109albii 1822 . . . 4 (∀𝑧(𝑥(𝐴𝐵)𝑧𝑥𝐶𝑧) ↔ ∀𝑧𝑦((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
11 alcom 2157 . . . 4 (∀𝑧𝑦((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ ∀𝑦𝑧((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
1210, 11bitri 275 . . 3 (∀𝑧(𝑥(𝐴𝐵)𝑧𝑥𝐶𝑧) ↔ ∀𝑦𝑧((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
1312albii 1822 . 2 (∀𝑥𝑧(𝑥(𝐴𝐵)𝑧𝑥𝐶𝑧) ↔ ∀𝑥𝑦𝑧((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
143, 13bitri 275 1 ((𝐴𝐵) ⊆ 𝐶 ↔ ∀𝑥𝑦𝑧((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wal 1540  wex 1782  wss 3914   class class class wbr 5109  ccom 5641  Rel wrel 5642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-11 2155  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-xp 5643  df-rel 5644  df-co 5646
This theorem is referenced by: (None)
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