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Theorem iinexd 43812
Description: The existence of an indexed union. 𝑥 is normally a free-variable parameter in 𝐵, which should be read 𝐵(𝑥). (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
iinexd.1 (𝜑𝐴 ≠ ∅)
iinexd.2 (𝜑 → ∀𝑥𝐴 𝐵𝐶)
Assertion
Ref Expression
iinexd (𝜑 𝑥𝐴 𝐵 ∈ V)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem iinexd
StepHypRef Expression
1 iinexd.1 . 2 (𝜑𝐴 ≠ ∅)
2 iinexd.2 . 2 (𝜑 → ∀𝑥𝐴 𝐵𝐶)
3 iinexg 5341 . 2 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵𝐶) → 𝑥𝐴 𝐵 ∈ V)
41, 2, 3syl2anc 584 1 (𝜑 𝑥𝐴 𝐵 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  wne 2940  wral 3061  Vcvv 3474  c0 4322   ciin 4998
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-in 3955  df-ss 3965  df-nul 4323  df-int 4951  df-iin 5000
This theorem is referenced by:  smfsuplem1  45517  smfinflem  45523  smflimsuplem1  45526  smflimsuplem2  45527  smflimsuplem3  45528  smflimsuplem4  45529  smflimsuplem5  45530  smflimsuplem7  45532
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