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Theorem cbvmpt21 38306
 Description: Rule to change the first bound variable in a maps-to function, using implicit substitution. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
cbvmpt21.1 𝑥𝐵
cbvmpt21.2 𝑧𝐵
cbvmpt21.3 𝑧𝐶
cbvmpt21.4 𝑥𝐸
cbvmpt21.5 (𝑥 = 𝑧𝐶 = 𝐸)
Assertion
Ref Expression
cbvmpt21 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧𝐴, 𝑦𝐵𝐸)
Distinct variable groups:   𝑥,𝐴,𝑧   𝑥,𝑦,𝑧
Allowed substitution hints:   𝐴(𝑦)   𝐵(𝑥,𝑦,𝑧)   𝐶(𝑥,𝑦,𝑧)   𝐸(𝑥,𝑦,𝑧)

Proof of Theorem cbvmpt21
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 nfv 1830 . . . . 5 𝑧 𝑥𝐴
2 cbvmpt21.2 . . . . . 6 𝑧𝐵
32nfcri 2745 . . . . 5 𝑧 𝑦𝐵
41, 3nfan 1816 . . . 4 𝑧(𝑥𝐴𝑦𝐵)
5 cbvmpt21.3 . . . . 5 𝑧𝐶
65nfeq2 2766 . . . 4 𝑧 𝑢 = 𝐶
74, 6nfan 1816 . . 3 𝑧((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶)
8 nfv 1830 . . . . 5 𝑥 𝑧𝐴
9 nfcv 2751 . . . . . 6 𝑥𝑦
10 cbvmpt21.1 . . . . . 6 𝑥𝐵
119, 10nfel 2763 . . . . 5 𝑥 𝑦𝐵
128, 11nfan 1816 . . . 4 𝑥(𝑧𝐴𝑦𝐵)
13 cbvmpt21.4 . . . . 5 𝑥𝐸
1413nfeq2 2766 . . . 4 𝑥 𝑢 = 𝐸
1512, 14nfan 1816 . . 3 𝑥((𝑧𝐴𝑦𝐵) ∧ 𝑢 = 𝐸)
16 eleq1 2676 . . . . 5 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
1716anbi1d 737 . . . 4 (𝑥 = 𝑧 → ((𝑥𝐴𝑦𝐵) ↔ (𝑧𝐴𝑦𝐵)))
18 cbvmpt21.5 . . . . 5 (𝑥 = 𝑧𝐶 = 𝐸)
1918eqeq2d 2620 . . . 4 (𝑥 = 𝑧 → (𝑢 = 𝐶𝑢 = 𝐸))
2017, 19anbi12d 743 . . 3 (𝑥 = 𝑧 → (((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶) ↔ ((𝑧𝐴𝑦𝐵) ∧ 𝑢 = 𝐸)))
217, 15, 20cbvoprab1 6625 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶)} = {⟨⟨𝑧, 𝑦⟩, 𝑢⟩ ∣ ((𝑧𝐴𝑦𝐵) ∧ 𝑢 = 𝐸)}
22 df-mpt2 6554 . 2 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶)}
23 df-mpt2 6554 . 2 (𝑧𝐴, 𝑦𝐵𝐸) = {⟨⟨𝑧, 𝑦⟩, 𝑢⟩ ∣ ((𝑧𝐴𝑦𝐵) ∧ 𝑢 = 𝐸)}
2421, 22, 233eqtr4i 2642 1 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧𝐴, 𝑦𝐵𝐸)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Ⅎwnfc 2738  {coprab 6550   ↦ cmpt2 6551 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-opab 4644  df-oprab 6553  df-mpt2 6554 This theorem is referenced by: (None)
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