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Theorem iuneq1 3661
Description: Equality theorem for indexed union. (Contributed by NM, 27-Jun-1998.)
Assertion
Ref Expression
iuneq1 (A = B x A 𝐶 = x B 𝐶)
Distinct variable groups:   x,A   x,B
Allowed substitution hint:   𝐶(x)

Proof of Theorem iuneq1
StepHypRef Expression
1 iunss1 3659 . . 3 (AB x A 𝐶 x B 𝐶)
2 iunss1 3659 . . 3 (BA x B 𝐶 x A 𝐶)
31, 2anim12i 321 . 2 ((AB BA) → ( x A 𝐶 x B 𝐶 x B 𝐶 x A 𝐶))
4 eqss 2954 . 2 (A = B ↔ (AB BA))
5 eqss 2954 . 2 ( x A 𝐶 = x B 𝐶 ↔ ( x A 𝐶 x B 𝐶 x B 𝐶 x A 𝐶))
63, 4, 53imtr4i 190 1 (A = B x A 𝐶 = x B 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242  wss 2911   ciun 3648
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-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  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-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-in 2918  df-ss 2925  df-iun 3650
This theorem is referenced by:  iuneq1d  3671  iununir  3729  iunsuc  4123  rdgisuc1  5911  rdg0  5914  oasuc  5983  omsuc  5990
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