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Theorem iinexd 44371
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 5332 . 2 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵𝐶) → 𝑥𝐴 𝐵 ∈ V)
41, 2, 3syl2anc 583 1 (𝜑 𝑥𝐴 𝐵 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  wne 2932  wral 3053  Vcvv 3466  c0 4315   ciin 4989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-in 3948  df-ss 3958  df-nul 4316  df-int 4942  df-iin 4991
This theorem is referenced by:  smfsuplem1  46073  smfinflem  46079  smflimsuplem1  46082  smflimsuplem2  46083  smflimsuplem3  46084  smflimsuplem4  46085  smflimsuplem5  46086  smflimsuplem7  46088
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