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Theorem iuneq1 3826
Description: Equality theorem for indexed union. (Contributed by NM, 27-Jun-1998.)
Assertion
Ref Expression
iuneq1 (𝐴 = 𝐵 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem iuneq1
StepHypRef Expression
1 iunss1 3824 . . 3 (𝐴𝐵 𝑥𝐴 𝐶 𝑥𝐵 𝐶)
2 iunss1 3824 . . 3 (𝐵𝐴 𝑥𝐵 𝐶 𝑥𝐴 𝐶)
31, 2anim12i 336 . 2 ((𝐴𝐵𝐵𝐴) → ( 𝑥𝐴 𝐶 𝑥𝐵 𝐶 𝑥𝐵 𝐶 𝑥𝐴 𝐶))
4 eqss 3112 . 2 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
5 eqss 3112 . 2 ( 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶 ↔ ( 𝑥𝐴 𝐶 𝑥𝐵 𝐶 𝑥𝐵 𝐶 𝑥𝐴 𝐶))
63, 4, 53imtr4i 200 1 (𝐴 = 𝐵 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1331  wss 3071   ciun 3813
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-in 3077  df-ss 3084  df-iun 3815
This theorem is referenced by:  iuneq1d  3836  iunxprg  3893  iununir  3896  iunsuc  4342  rdgisuc1  6281  rdg0  6284  oasuc  6360  omsuc  6368  iunfidisj  6834  fsum2d  11211  fsumiun  11253  iuncld  12294
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