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Theorem iuneq1 3640
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 3638 . . 3 (AB x A 𝐶 x B 𝐶)
2 iunss1 3638 . . 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 2933 . 2 (A = B ↔ (AB BA))
5 eqss 2933 . 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 1226  wss 2890   ciun 3627
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-in 2897  df-ss 2904  df-iun 3629
This theorem is referenced by:  iuneq1d  3650  iununir  3708  iunsuc  4102  rdgi0g  5882  rdgisuc1  5887  rdg0  5891  oasuc  5955  omsuc  5962
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