New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  mpt2eq123dv GIF version

Theorem mpt2eq123dv 5663
 Description: An equality deduction for the maps to notation. (Contributed by set.mm contributors, 12-Sep-2011.)
Hypotheses
Ref Expression
mpt2eq123dv.1 (φA = D)
mpt2eq123dv.2 (φB = E)
mpt2eq123dv.3 (φC = F)
Assertion
Ref Expression
mpt2eq123dv (φ → (x A, y B C) = (x D, y E F))
Distinct variable groups:   φ,x   φ,y
Allowed substitution hints:   A(x,y)   B(x,y)   C(x,y)   D(x,y)   E(x,y)   F(x,y)

Proof of Theorem mpt2eq123dv
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 mpt2eq123dv.1 . . . . . 6 (φA = D)
21eleq2d 2420 . . . . 5 (φ → (x Ax D))
3 mpt2eq123dv.2 . . . . . 6 (φB = E)
43eleq2d 2420 . . . . 5 (φ → (y By E))
52, 4anbi12d 691 . . . 4 (φ → ((x A y B) ↔ (x D y E)))
6 mpt2eq123dv.3 . . . . 5 (φC = F)
76eqeq2d 2364 . . . 4 (φ → (z = Cz = F))
85, 7anbi12d 691 . . 3 (φ → (((x A y B) z = C) ↔ ((x D y E) z = F)))
98oprabbidv 5564 . 2 (φ → {x, y, z ((x A y B) z = C)} = {x, y, z ((x D y E) z = F)})
10 df-mpt2 5654 . 2 (x A, y B C) = {x, y, z ((x A y B) z = C)}
11 df-mpt2 5654 . 2 (x D, y E F) = {x, y, z ((x D y E) z = F)}
129, 10, 113eqtr4g 2410 1 (φ → (x A, y B C) = (x D, y E F))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {coprab 5527   ↦ cmpt2 5653 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-oprab 5528  df-mpt2 5654 This theorem is referenced by:  mpt2eq123i  5664
 Copyright terms: Public domain W3C validator