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Theorem iuneq2 4503
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 4502 . . 3 (∀𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 𝑥𝐴 𝐶)
2 ss2iun 4502 . . 3 (∀𝑥𝐴 𝐶𝐵 𝑥𝐴 𝐶 𝑥𝐴 𝐵)
31, 2anim12i 589 . 2 ((∀𝑥𝐴 𝐵𝐶 ∧ ∀𝑥𝐴 𝐶𝐵) → ( 𝑥𝐴 𝐵 𝑥𝐴 𝐶 𝑥𝐴 𝐶 𝑥𝐴 𝐵))
4 eqss 3598 . . . 4 (𝐵 = 𝐶 ↔ (𝐵𝐶𝐶𝐵))
54ralbii 2974 . . 3 (∀𝑥𝐴 𝐵 = 𝐶 ↔ ∀𝑥𝐴 (𝐵𝐶𝐶𝐵))
6 r19.26 3057 . . 3 (∀𝑥𝐴 (𝐵𝐶𝐶𝐵) ↔ (∀𝑥𝐴 𝐵𝐶 ∧ ∀𝑥𝐴 𝐶𝐵))
75, 6bitri 264 . 2 (∀𝑥𝐴 𝐵 = 𝐶 ↔ (∀𝑥𝐴 𝐵𝐶 ∧ ∀𝑥𝐴 𝐶𝐵))
8 eqss 3598 . 2 ( 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶 ↔ ( 𝑥𝐴 𝐵 𝑥𝐴 𝐶 𝑥𝐴 𝐶 𝑥𝐴 𝐵))
93, 7, 83imtr4i 281 1 (∀𝑥𝐴 𝐵 = 𝐶 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wral 2907  wss 3555   ciun 4485
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-v 3188  df-in 3562  df-ss 3569  df-iun 4487
This theorem is referenced by:  iuneq2i  4505  iuneq2dv  4508  iunxdif3  4572  oa0r  7563  om0r  7564  om1r  7568  oe1m  7570  oaass  7586  oarec  7587  omass  7605  oeoalem  7621  oeoelem  7623  cardiun  8752  kmlem11  8926  iuncld  20759  comppfsc  21245  istotbnd3  33199  sstotbnd  33203  heibor  33249  iuneq12f  33601  cnvtrclfv  37494  iuneq2df  38694
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