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Theorem iinssd 39628
 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 3665 . . . 4 (𝑥 = 𝑋 → (𝐵𝐶𝐷𝐶))
54rspcev 3340 . . 3 ((𝑋𝐴𝐷𝐶) → ∃𝑥𝐴 𝐵𝐶)
61, 2, 5syl2anc 694 . 2 (𝜑 → ∃𝑥𝐴 𝐵𝐶)
7 iinss 4603 . 2 (∃𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵𝐶)
86, 7syl 17 1 (𝜑 𝑥𝐴 𝐵𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1523   ∈ wcel 2030  ∃wrex 2942   ⊆ wss 3607  ∩ ciin 4553 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233  df-in 3614  df-ss 3621  df-iin 4555 This theorem is referenced by:  smfsuplem3  41340  smflimsuplem1  41347
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