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Theorem iuneq1 3925
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 3923 . . 3 (𝐴𝐵 𝑥𝐴 𝐶 𝑥𝐵 𝐶)
2 iunss1 3923 . . 3 (𝐵𝐴 𝑥𝐵 𝐶 𝑥𝐴 𝐶)
31, 2anim12i 338 . 2 ((𝐴𝐵𝐵𝐴) → ( 𝑥𝐴 𝐶 𝑥𝐵 𝐶 𝑥𝐵 𝐶 𝑥𝐴 𝐶))
4 eqss 3194 . 2 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
5 eqss 3194 . 2 ( 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶 ↔ ( 𝑥𝐴 𝐶 𝑥𝐵 𝐶 𝑥𝐵 𝐶 𝑥𝐴 𝐶))
63, 4, 53imtr4i 201 1 (𝐴 = 𝐵 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  wss 3153   ciun 3912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-in 3159  df-ss 3166  df-iun 3914
This theorem is referenced by:  iuneq1d  3935  iunxprg  3993  iununir  3996  iunsuc  4451  rdgisuc1  6437  rdg0  6440  oasuc  6517  omsuc  6525  iunfidisj  7005  fsum2d  11578  fsumiun  11620  fprod2d  11766  iuncld  14283
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