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Theorem invdisjrab 4562
Description: The restricted class abstractions {𝑥𝐵𝐶 = 𝑦} for distinct 𝑦𝐴 are disjoint. (Contributed by AV, 6-May-2020.)
Assertion
Ref Expression
invdisjrab Disj 𝑦𝐴 {𝑥𝐵𝐶 = 𝑦}
Distinct variable groups:   𝑥,𝐵   𝑦,𝐶   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑦)   𝐶(𝑥)

Proof of Theorem invdisjrab
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfcv 2746 . . . . . 6 𝑥𝑧
2 nfcv 2746 . . . . . 6 𝑥𝐵
3 nfcsb1v 3510 . . . . . . 7 𝑥𝑧 / 𝑥𝐶
43nfeq1 2759 . . . . . 6 𝑥𝑧 / 𝑥𝐶 = 𝑦
5 csbeq1a 3503 . . . . . . 7 (𝑥 = 𝑧𝐶 = 𝑧 / 𝑥𝐶)
65eqeq1d 2607 . . . . . 6 (𝑥 = 𝑧 → (𝐶 = 𝑦𝑧 / 𝑥𝐶 = 𝑦))
71, 2, 4, 6elrabf 3324 . . . . 5 (𝑧 ∈ {𝑥𝐵𝐶 = 𝑦} ↔ (𝑧𝐵𝑧 / 𝑥𝐶 = 𝑦))
8 ax-1 6 . . . . 5 (𝑧 / 𝑥𝐶 = 𝑦 → (𝑦𝐴𝑧 / 𝑥𝐶 = 𝑦))
97, 8simplbiim 656 . . . 4 (𝑧 ∈ {𝑥𝐵𝐶 = 𝑦} → (𝑦𝐴𝑧 / 𝑥𝐶 = 𝑦))
109impcom 444 . . 3 ((𝑦𝐴𝑧 ∈ {𝑥𝐵𝐶 = 𝑦}) → 𝑧 / 𝑥𝐶 = 𝑦)
1110rgen2 2953 . 2 𝑦𝐴𝑧 ∈ {𝑥𝐵𝐶 = 𝑦}𝑧 / 𝑥𝐶 = 𝑦
12 invdisj 4561 . 2 (∀𝑦𝐴𝑧 ∈ {𝑥𝐵𝐶 = 𝑦}𝑧 / 𝑥𝐶 = 𝑦Disj 𝑦𝐴 {𝑥𝐵𝐶 = 𝑦})
1311, 12ax-mp 5 1 Disj 𝑦𝐴 {𝑥𝐵𝐶 = 𝑦}
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1975  wral 2891  {crab 2895  csb 3494  Disj wdisj 4543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ral 2896  df-rmo 2899  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-disj 4544
This theorem is referenced by:  disjxwrd  13249  disjwrdpfx  40072
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