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Theorem iinssdf 41703
 Description: Subset implication for an indexed intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
iinssdf.a 𝑥𝐴
iinssdf.n 𝑥𝑋
iinssdf.c 𝑥𝐶
iinssdf.d 𝑥𝐷
iinssdf.x (𝜑𝑋𝐴)
iinssdf.b (𝑥 = 𝑋𝐵 = 𝐷)
iinssdf.s (𝜑𝐷𝐶)
Assertion
Ref Expression
iinssdf (𝜑 𝑥𝐴 𝐵𝐶)

Proof of Theorem iinssdf
StepHypRef Expression
1 iinssdf.x . . 3 (𝜑𝑋𝐴)
2 iinssdf.s . . 3 (𝜑𝐷𝐶)
3 iinssdf.d . . . . 5 𝑥𝐷
4 iinssdf.c . . . . 5 𝑥𝐶
53, 4nfss 3945 . . . 4 𝑥 𝐷𝐶
6 iinssdf.n . . . 4 𝑥𝑋
7 iinssdf.a . . . 4 𝑥𝐴
8 iinssdf.b . . . . 5 (𝑥 = 𝑋𝐵 = 𝐷)
98sseq1d 3984 . . . 4 (𝑥 = 𝑋 → (𝐵𝐶𝐷𝐶))
105, 6, 7, 9rspcef 41630 . . 3 ((𝑋𝐴𝐷𝐶) → ∃𝑥𝐴 𝐵𝐶)
111, 2, 10syl2anc 587 . 2 (𝜑 → ∃𝑥𝐴 𝐵𝐶)
124iinssf 41702 . 2 (∃𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵𝐶)
1311, 12syl 17 1 (𝜑 𝑥𝐴 𝐵𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2115  Ⅎwnfc 2962  ∃wrex 3134   ⊆ wss 3919  ∩ ciin 4906 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796 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 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-v 3482  df-in 3926  df-ss 3936  df-iin 4908 This theorem is referenced by: (None)
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