Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  iinssiin Structured version   Visualization version   GIF version

Theorem iinssiin 40121
 Description: Subset implication for an indexed intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
iinssiin.1 𝑥𝜑
iinssiin.2 ((𝜑𝑥𝐴) → 𝐵𝐶)
Assertion
Ref Expression
iinssiin (𝜑 𝑥𝐴 𝐵 𝑥𝐴 𝐶)

Proof of Theorem iinssiin
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 iinssiin.1 . . . . . 6 𝑥𝜑
2 nfcv 2969 . . . . . . 7 𝑥𝑦
3 nfii1 4773 . . . . . . 7 𝑥 𝑥𝐴 𝐵
42, 3nfel 2982 . . . . . 6 𝑥 𝑦 𝑥𝐴 𝐵
51, 4nfan 2002 . . . . 5 𝑥(𝜑𝑦 𝑥𝐴 𝐵)
6 iinssiin.2 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐵𝐶)
76adantlr 706 . . . . . . 7 (((𝜑𝑦 𝑥𝐴 𝐵) ∧ 𝑥𝐴) → 𝐵𝐶)
8 eliinid 40104 . . . . . . . 8 ((𝑦 𝑥𝐴 𝐵𝑥𝐴) → 𝑦𝐵)
98adantll 705 . . . . . . 7 (((𝜑𝑦 𝑥𝐴 𝐵) ∧ 𝑥𝐴) → 𝑦𝐵)
107, 9sseldd 3828 . . . . . 6 (((𝜑𝑦 𝑥𝐴 𝐵) ∧ 𝑥𝐴) → 𝑦𝐶)
1110ex 403 . . . . 5 ((𝜑𝑦 𝑥𝐴 𝐵) → (𝑥𝐴𝑦𝐶))
125, 11ralrimi 3166 . . . 4 ((𝜑𝑦 𝑥𝐴 𝐵) → ∀𝑥𝐴 𝑦𝐶)
13 vex 3417 . . . . 5 𝑦 ∈ V
14 eliin 4747 . . . . 5 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐶 ↔ ∀𝑥𝐴 𝑦𝐶))
1513, 14ax-mp 5 . . . 4 (𝑦 𝑥𝐴 𝐶 ↔ ∀𝑥𝐴 𝑦𝐶)
1612, 15sylibr 226 . . 3 ((𝜑𝑦 𝑥𝐴 𝐵) → 𝑦 𝑥𝐴 𝐶)
1716ralrimiva 3175 . 2 (𝜑 → ∀𝑦 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐶)
18 dfss3 3816 . 2 ( 𝑥𝐴 𝐵 𝑥𝐴 𝐶 ↔ ∀𝑦 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐶)
1917, 18sylibr 226 1 (𝜑 𝑥𝐴 𝐵 𝑥𝐴 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386  Ⅎwnf 1882   ∈ wcel 2164  ∀wral 3117  Vcvv 3414   ⊆ wss 3798  ∩ ciin 4743 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-13 2389  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-v 3416  df-in 3805  df-ss 3812  df-iin 4745 This theorem is referenced by: (None)
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