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Theorem iineq1d 45029
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 5013 . 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 1536   ciin 4996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-ral 3059  df-rex 3068  df-iin 4998
This theorem is referenced by:  smflimlem2  46727  smflimlem3  46728  smflimlem4  46729  smflim2  46761  smflimsuplem1  46775  smflimsuplem7  46781  smflimsup  46783  smfliminf  46786
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