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Theorem iinssd 40128
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 3857 . . . 4 (𝑥 = 𝑋 → (𝐵𝐶𝐷𝐶))
54rspcev 3526 . . 3 ((𝑋𝐴𝐷𝐶) → ∃𝑥𝐴 𝐵𝐶)
61, 2, 5syl2anc 581 . 2 (𝜑 → ∃𝑥𝐴 𝐵𝐶)
7 iinss 4791 . 2 (∃𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵𝐶)
86, 7syl 17 1 (𝜑 𝑥𝐴 𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1658  wcel 2166  wrex 3118  wss 3798   ciin 4741
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-ext 2803
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-v 3416  df-in 3805  df-ss 3812  df-iin 4743
This theorem is referenced by:  smfsuplem3  41813  smflimsuplem1  41820
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