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Theorem mpt2eq3dva 5669
 Description: Slightly more general equality inference for the maps to notation. (Contributed by set.mm contributors, 17-Oct-2013.) (Revised by set.mm contributors, 16-Dec-2013.)
Hypothesis
Ref Expression
mpt2eq3dva.1 ((φ x A y B) → C = D)
Assertion
Ref Expression
mpt2eq3dva (φ → (x A, y B C) = (x A, y B D))
Distinct variable groups:   φ,x   φ,y
Allowed substitution hints:   A(x,y)   B(x,y)   C(x,y)   D(x,y)

Proof of Theorem mpt2eq3dva
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 mpt2eq3dva.1 . . . . . 6 ((φ x A y B) → C = D)
213expb 1152 . . . . 5 ((φ (x A y B)) → C = D)
32eqeq2d 2364 . . . 4 ((φ (x A y B)) → (z = Cz = D))
43pm5.32da 622 . . 3 (φ → (((x A y B) z = C) ↔ ((x A y B) z = D)))
54oprabbidv 5564 . 2 (φ → {x, y, z ((x A y B) z = C)} = {x, y, z ((x A y B) z = D)})
6 df-mpt2 5654 . 2 (x A, y B C) = {x, y, z ((x A y B) z = C)}
7 df-mpt2 5654 . 2 (x A, y B D) = {x, y, z ((x A y B) z = D)}
85, 6, 73eqtr4g 2410 1 (φ → (x A, y B C) = (x A, y B D))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   ∧ w3a 934   = 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-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-oprab 5528  df-mpt2 5654 This theorem is referenced by:  mpt2eq3ia  5670
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