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Theorem iinssd 44376
Description: Subset implication for an indexed intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
iinssd.1 (𝜑𝑋𝐴)
iinssd.2 (𝑥 = 𝑋𝐵 = 𝐷)
iinssd.3 (𝜑𝐷𝐶)
Assertion
Ref Expression
iinssd (𝜑 𝑥𝐴 𝐵𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷   𝑥,𝑋
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem iinssd
StepHypRef Expression
1 iinssd.1 . . 3 (𝜑𝑋𝐴)
2 iinssd.3 . . 3 (𝜑𝐷𝐶)
3 iinssd.2 . . . . 5 (𝑥 = 𝑋𝐵 = 𝐷)
43sseq1d 4008 . . . 4 (𝑥 = 𝑋 → (𝐵𝐶𝐷𝐶))
54rspcev 3606 . . 3 ((𝑋𝐴𝐷𝐶) → ∃𝑥𝐴 𝐵𝐶)
61, 2, 5syl2anc 583 . 2 (𝜑 → ∃𝑥𝐴 𝐵𝐶)
7 iinss 5052 . 2 (∃𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵𝐶)
86, 7syl 17 1 (𝜑 𝑥𝐴 𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  wrex 3064  wss 3943   ciin 4991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-v 3470  df-in 3950  df-ss 3960  df-iin 4993
This theorem is referenced by:  smfsuplem3  46082  smflimsuplem1  46089
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