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Theorem mpt2eq3dva 5695
Description: Slightly more general equality inference for the maps-to notation. (Contributed by NM, 17-Oct-2013.)
Hypothesis
Ref Expression
mpt2eq3dva.1 ((𝜑𝑥𝐴𝑦𝐵) → 𝐶 = 𝐷)
Assertion
Ref Expression
mpt2eq3dva (𝜑 → (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐴, 𝑦𝐵𝐷))
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 ((𝜑𝑥𝐴𝑦𝐵) → 𝐶 = 𝐷)
213expb 1144 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → 𝐶 = 𝐷)
32eqeq2d 2099 . . . 4 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝑧 = 𝐶𝑧 = 𝐷))
43pm5.32da 440 . . 3 (𝜑 → (((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐷)))
54oprabbidv 5685 . 2 (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐷)})
6 df-mpt2 5639 . 2 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
7 df-mpt2 5639 . 2 (𝑥𝐴, 𝑦𝐵𝐷) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐷)}
85, 6, 73eqtr4g 2145 1 (𝜑 → (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐴, 𝑦𝐵𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  w3a 924   = wceq 1289  wcel 1438  {coprab 5635  cmpt2 5636
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-oprab 5638  df-mpt2 5639
This theorem is referenced by:  mpt2eq3ia  5696  ofeq  5840  fmpt2co  5963  mapxpen  6544  iseqeq2  9823  iseqeq3  9824  iseqval  9835
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