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Theorem iuneq2 3698
Description: Equality theorem for indexed union. (Contributed by NM, 22-Oct-2003.)
Assertion
Ref Expression
iuneq2 (∀𝑥𝐴 𝐵 = 𝐶 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)

Proof of Theorem iuneq2
StepHypRef Expression
1 ss2iun 3697 . . 3 (∀𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 𝑥𝐴 𝐶)
2 ss2iun 3697 . . 3 (∀𝑥𝐴 𝐶𝐵 𝑥𝐴 𝐶 𝑥𝐴 𝐵)
31, 2anim12i 325 . 2 ((∀𝑥𝐴 𝐵𝐶 ∧ ∀𝑥𝐴 𝐶𝐵) → ( 𝑥𝐴 𝐵 𝑥𝐴 𝐶 𝑥𝐴 𝐶 𝑥𝐴 𝐵))
4 eqss 2985 . . . 4 (𝐵 = 𝐶 ↔ (𝐵𝐶𝐶𝐵))
54ralbii 2345 . . 3 (∀𝑥𝐴 𝐵 = 𝐶 ↔ ∀𝑥𝐴 (𝐵𝐶𝐶𝐵))
6 r19.26 2456 . . 3 (∀𝑥𝐴 (𝐵𝐶𝐶𝐵) ↔ (∀𝑥𝐴 𝐵𝐶 ∧ ∀𝑥𝐴 𝐶𝐵))
75, 6bitri 177 . 2 (∀𝑥𝐴 𝐵 = 𝐶 ↔ (∀𝑥𝐴 𝐵𝐶 ∧ ∀𝑥𝐴 𝐶𝐵))
8 eqss 2985 . 2 ( 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶 ↔ ( 𝑥𝐴 𝐵 𝑥𝐴 𝐶 𝑥𝐴 𝐶 𝑥𝐴 𝐵))
93, 7, 83imtr4i 194 1 (∀𝑥𝐴 𝐵 = 𝐶 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1257  wral 2321  wss 2942   ciun 3682
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036
This theorem depends on definitions:  df-bi 114  df-tru 1260  df-nf 1364  df-sb 1660  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ral 2326  df-rex 2327  df-v 2574  df-in 2949  df-ss 2956  df-iun 3684
This theorem is referenced by:  iuneq2i  3700  iuneq2dv  3703  dfmptg  5367
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