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Theorem iuneq2dv 3990
 Description: Equality deduction for indexed union. (Contributed by NM, 3-Aug-2004.)
Hypothesis
Ref Expression
iuneq2dv.1 ((φ x A) → B = C)
Assertion
Ref Expression
iuneq2dv (φx A B = x A C)
Distinct variable group:   φ,x
Allowed substitution hints:   A(x)   B(x)   C(x)

Proof of Theorem iuneq2dv
StepHypRef Expression
1 iuneq2dv.1 . . 3 ((φ x A) → B = C)
21ralrimiva 2697 . 2 (φx A B = C)
3 iuneq2 3985 . 2 (x A B = Cx A B = x A C)
42, 3syl 15 1 (φx A B = x A C)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2614  ∪ciun 3969 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-iun 3971 This theorem is referenced by:  iuneq12d  3993  iuneq2d  3994
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