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Theorem iinssd 41702
 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 3973 . . . 4 (𝑥 = 𝑋 → (𝐵𝐶𝐷𝐶))
54rspcev 3598 . . 3 ((𝑋𝐴𝐷𝐶) → ∃𝑥𝐴 𝐵𝐶)
61, 2, 5syl2anc 587 . 2 (𝜑 → ∃𝑥𝐴 𝐵𝐶)
7 iinss 4955 . 2 (∃𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵𝐶)
86, 7syl 17 1 (𝜑 𝑥𝐴 𝐵𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2114  ∃wrex 3131   ⊆ wss 3908  ∩ ciin 4895 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-v 3471  df-in 3915  df-ss 3925  df-iin 4897 This theorem is referenced by:  smfsuplem3  43383  smflimsuplem1  43390
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