New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  iuneq12d GIF version

Theorem iuneq12d 3993
 Description: Equality deduction for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015.)
Hypotheses
Ref Expression
iuneq1d.1 (φA = B)
iuneq12d.2 (φC = D)
Assertion
Ref Expression
iuneq12d (φx A C = x B D)
Distinct variable groups:   x,A   x,B   φ,x
Allowed substitution hints:   C(x)   D(x)

Proof of Theorem iuneq12d
StepHypRef Expression
1 iuneq1d.1 . . 3 (φA = B)
21iuneq1d 3992 . 2 (φx A C = x B C)
3 iuneq12d.2 . . . 4 (φC = D)
43adantr 451 . . 3 ((φ x B) → C = D)
54iuneq2dv 3990 . 2 (φx B C = x B D)
62, 5eqtrd 2385 1 (φx A C = x B D)
 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: (None)
 Copyright terms: Public domain W3C validator