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Theorem iineq2 3665
 Description: Equality theorem for indexed intersection. (Contributed by NM, 22-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
iineq2 (x A B = 𝐶 x A B = x A 𝐶)

Proof of Theorem iineq2
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 eleq2 2098 . . . . 5 (B = 𝐶 → (y By 𝐶))
21ralimi 2378 . . . 4 (x A B = 𝐶x A (y By 𝐶))
3 ralbi 2439 . . . 4 (x A (y By 𝐶) → (x A y Bx A y 𝐶))
42, 3syl 14 . . 3 (x A B = 𝐶 → (x A y Bx A y 𝐶))
54abbidv 2152 . 2 (x A B = 𝐶 → {yx A y B} = {yx A y 𝐶})
6 df-iin 3651 . 2 x A B = {yx A y B}
7 df-iin 3651 . 2 x A 𝐶 = {yx A y 𝐶}
85, 6, 73eqtr4g 2094 1 (x A B = 𝐶 x A B = x A 𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1242   ∈ wcel 1390  {cab 2023  ∀wral 2300  ∩ ciin 3649 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-ral 2305  df-iin 3651 This theorem is referenced by:  iineq2i  3667  iineq2d  3668
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