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Theorem iineq1d 42640
Description: Equality theorem for indexed intersection. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
Hypothesis
Ref Expression
iineq1d.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
iineq1d (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝐶(𝑥)

Proof of Theorem iineq1d
StepHypRef Expression
1 iineq1d.1 . 2 (𝜑𝐴 = 𝐵)
2 iineq1 4941 . 2 (𝐴 = 𝐵 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)
31, 2syl 17 1 (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539   ciin 4925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-ral 3069  df-iin 4927
This theorem is referenced by:  iineq12dv  42656  smflimlem2  44307  smflimlem3  44308  smflimlem4  44309  smflim2  44339  smflimsuplem1  44353  smflimsuplem7  44359  smflimsup  44361  smfliminf  44364
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