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Theorem cbvmpt21 39592
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 1883 . . . . 5 𝑧 𝑥𝐴
2 cbvmpt21.2 . . . . . 6 𝑧𝐵
32nfcri 2787 . . . . 5 𝑧 𝑦𝐵
41, 3nfan 1868 . . . 4 𝑧(𝑥𝐴𝑦𝐵)
5 cbvmpt21.3 . . . . 5 𝑧𝐶
65nfeq2 2809 . . . 4 𝑧 𝑢 = 𝐶
74, 6nfan 1868 . . 3 𝑧((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶)
8 nfv 1883 . . . . 5 𝑥 𝑧𝐴
9 nfcv 2793 . . . . . 6 𝑥𝑦
10 cbvmpt21.1 . . . . . 6 𝑥𝐵
119, 10nfel 2806 . . . . 5 𝑥 𝑦𝐵
128, 11nfan 1868 . . . 4 𝑥(𝑧𝐴𝑦𝐵)
13 cbvmpt21.4 . . . . 5 𝑥𝐸
1413nfeq2 2809 . . . 4 𝑥 𝑢 = 𝐸
1512, 14nfan 1868 . . 3 𝑥((𝑧𝐴𝑦𝐵) ∧ 𝑢 = 𝐸)
16 eleq1 2718 . . . . 5 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
1716anbi1d 741 . . . 4 (𝑥 = 𝑧 → ((𝑥𝐴𝑦𝐵) ↔ (𝑧𝐴𝑦𝐵)))
18 cbvmpt21.5 . . . . 5 (𝑥 = 𝑧𝐶 = 𝐸)
1918eqeq2d 2661 . . . 4 (𝑥 = 𝑧 → (𝑢 = 𝐶𝑢 = 𝐸))
2017, 19anbi12d 747 . . 3 (𝑥 = 𝑧 → (((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶) ↔ ((𝑧𝐴𝑦𝐵) ∧ 𝑢 = 𝐸)))
217, 15, 20cbvoprab1 6769 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶)} = {⟨⟨𝑧, 𝑦⟩, 𝑢⟩ ∣ ((𝑧𝐴𝑦𝐵) ∧ 𝑢 = 𝐸)}
22 df-mpt2 6695 . 2 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶)}
23 df-mpt2 6695 . 2 (𝑧𝐴, 𝑦𝐵𝐸) = {⟨⟨𝑧, 𝑦⟩, 𝑢⟩ ∣ ((𝑧𝐴𝑦𝐵) ∧ 𝑢 = 𝐸)}
2421, 22, 233eqtr4i 2683 1 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧𝐴, 𝑦𝐵𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wnfc 2780  {coprab 6691  cmpt2 6692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-opab 4746  df-oprab 6694  df-mpt2 6695
This theorem is referenced by: (None)
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