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Theorem iineq2i 4674
 Description: Equality inference for indexed intersection. (Contributed by NM, 22-Oct-2003.)
Hypothesis
Ref Expression
iuneq2i.1 (𝑥𝐴𝐵 = 𝐶)
Assertion
Ref Expression
iineq2i 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶

Proof of Theorem iineq2i
StepHypRef Expression
1 iineq2 4672 . 2 (∀𝑥𝐴 𝐵 = 𝐶 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)
2 iuneq2i.1 . 2 (𝑥𝐴𝐵 = 𝐶)
31, 2mprg 3075 1 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1631   ∈ wcel 2145  ∩ ciin 4655 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-ral 3066  df-iin 4657 This theorem is referenced by:  iinrab  4716  iinin1  4725  diaintclN  36868  dibintclN  36977  dihintcl  37154  imaiinfv  37782  smflimlem3  41501
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