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Theorem iinexd 45073
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 5354 . 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 2938  wral 3059  Vcvv 3478  c0 4339   ciin 4997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-in 3970  df-ss 3980  df-nul 4340  df-int 4952  df-iin 4999
This theorem is referenced by:  smfsuplem1  46767  smfinflem  46773  smflimsuplem1  46776  smflimsuplem2  46777  smflimsuplem3  46778  smflimsuplem4  46779  smflimsuplem5  46780  smflimsuplem7  46782
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