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Theorem mpt2eq12 5565
 Description: An equality theorem for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
Assertion
Ref Expression
mpt2eq12 ((𝐴 = 𝐶𝐵 = 𝐷) → (𝑥𝐴, 𝑦𝐵𝐸) = (𝑥𝐶, 𝑦𝐷𝐸))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝐷,𝑦
Allowed substitution hints:   𝐸(𝑥,𝑦)

Proof of Theorem mpt2eq12
StepHypRef Expression
1 eqid 2040 . . . . 5 𝐸 = 𝐸
21rgenw 2376 . . . 4 𝑦𝐵 𝐸 = 𝐸
32jctr 298 . . 3 (𝐵 = 𝐷 → (𝐵 = 𝐷 ∧ ∀𝑦𝐵 𝐸 = 𝐸))
43ralrimivw 2393 . 2 (𝐵 = 𝐷 → ∀𝑥𝐴 (𝐵 = 𝐷 ∧ ∀𝑦𝐵 𝐸 = 𝐸))
5 mpt2eq123 5564 . 2 ((𝐴 = 𝐶 ∧ ∀𝑥𝐴 (𝐵 = 𝐷 ∧ ∀𝑦𝐵 𝐸 = 𝐸)) → (𝑥𝐴, 𝑦𝐵𝐸) = (𝑥𝐶, 𝑦𝐷𝐸))
64, 5sylan2 270 1 ((𝐴 = 𝐶𝐵 = 𝐷) → (𝑥𝐴, 𝑦𝐵𝐸) = (𝑥𝐶, 𝑦𝐷𝐸))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243  ∀wral 2306   ↦ cmpt2 5514 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-oprab 5516  df-mpt2 5517 This theorem is referenced by:  iseqeq1  9214  iseqeq4  9217
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