New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  iuneq2d GIF version

Theorem iuneq2d 3994
 Description: Equality deduction for indexed union. (Contributed by Drahflow, 22-Oct-2015.)
Hypothesis
Ref Expression
iuneq2d.2 (φB = C)
Assertion
Ref Expression
iuneq2d (φx A B = x A C)
Distinct variable groups:   φ,x   x,A
Allowed substitution hints:   B(x)   C(x)

Proof of Theorem iuneq2d
StepHypRef Expression
1 iuneq2d.2 . . 3 (φB = C)
21adantr 451 . 2 ((φ x A) → B = C)
32iuneq2dv 3990 1 (φx A B = x A C)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  ∪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:  iununi  4050
 Copyright terms: Public domain W3C validator