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Theorem intabssd 37394
Description: When for each element 𝑦 there is a subset 𝐴 which may substituted for 𝑥 such that 𝑦 satisfying 𝜒 implies 𝑥 satisfies 𝜓 then the intersection of all 𝑥 that satisfy 𝜓 is a subclass the intersection of all 𝑦 that satisfy 𝜒. (Contributed by RP, 17-Oct-2020.)
Hypotheses
Ref Expression
intabssd.ex (𝜑𝐴𝑉)
intabssd.sub ((𝜑𝑥 = 𝐴) → (𝜒𝜓))
intabssd.ss (𝜑𝐴𝑦)
Assertion
Ref Expression
intabssd (𝜑 {𝑥𝜓} ⊆ {𝑦𝜒})
Distinct variable groups:   𝜒,𝑥   𝜓,𝑦   𝑥,𝑦,𝜑   𝑥,𝐴
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)   𝐴(𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem intabssd
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 intabssd.ex . . . . 5 (𝜑𝐴𝑉)
2 intabssd.sub . . . . . 6 ((𝜑𝑥 = 𝐴) → (𝜒𝜓))
3 eleq2 2687 . . . . . . . 8 (𝑥 = 𝐴 → (𝑧𝑥𝑧𝐴))
43biimpd 219 . . . . . . 7 (𝑥 = 𝐴 → (𝑧𝑥𝑧𝐴))
5 intabssd.ss . . . . . . . 8 (𝜑𝐴𝑦)
65sseld 3582 . . . . . . 7 (𝜑 → (𝑧𝐴𝑧𝑦))
74, 6sylan9r 689 . . . . . 6 ((𝜑𝑥 = 𝐴) → (𝑧𝑥𝑧𝑦))
82, 7imim12d 81 . . . . 5 ((𝜑𝑥 = 𝐴) → ((𝜓𝑧𝑥) → (𝜒𝑧𝑦)))
91, 8spcimdv 3276 . . . 4 (𝜑 → (∀𝑥(𝜓𝑧𝑥) → (𝜒𝑧𝑦)))
109alrimdv 1854 . . 3 (𝜑 → (∀𝑥(𝜓𝑧𝑥) → ∀𝑦(𝜒𝑧𝑦)))
11 vex 3189 . . . 4 𝑧 ∈ V
1211elintab 4452 . . 3 (𝑧 {𝑥𝜓} ↔ ∀𝑥(𝜓𝑧𝑥))
1311elintab 4452 . . 3 (𝑧 {𝑦𝜒} ↔ ∀𝑦(𝜒𝑧𝑦))
1410, 12, 133imtr4g 285 . 2 (𝜑 → (𝑧 {𝑥𝜓} → 𝑧 {𝑦𝜒}))
1514ssrdv 3589 1 (𝜑 {𝑥𝜓} ⊆ {𝑦𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1478   = wceq 1480  wcel 1987  {cab 2607  wss 3555   cint 4440
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188  df-in 3562  df-ss 3569  df-int 4441
This theorem is referenced by:  clcnvlem  37408
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