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Theorem iineq1d 38295
 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 4471 . 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 1475  ∩ ciin 4456 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-iin 4458 This theorem is referenced by:  iineq12dv  38319  smflimlem2  39658  smflimlem3  39659  smflimlem4  39660
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