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Theorem iineq2i 5013
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 5011 . 2 (∀𝑥𝐴 𝐵 = 𝐶 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)
2 iuneq2i.1 . 2 (𝑥𝐴𝐵 = 𝐶)
31, 2mprg 3066 1 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107   ciin 4991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-iin 4993
This theorem is referenced by:  iinrab  5068  iinin1  5078  diaintclN  41061  dibintclN  41170  dihintcl  41347  imaiinfv  42709  smflimlem3  46793
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