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Theorem iineq1d 40021
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 4726 . 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 1653   ciin 4712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2778
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ral 3095  df-iin 4714
This theorem is referenced by:  iineq12dv  40042  smflimlem2  41721  smflimlem3  41722  smflimlem4  41723  smflim2  41753  smflimsuplem1  41767  smflimsuplem7  41773  smflimsup  41775  smfliminf  41778
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