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Theorem invdisjrab 4671
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 2793 . . . . . 6 𝑥𝑧
2 nfcv 2793 . . . . . 6 𝑥𝐵
3 nfcsb1v 3582 . . . . . . 7 𝑥𝑧 / 𝑥𝐶
43nfeq1 2807 . . . . . 6 𝑥𝑧 / 𝑥𝐶 = 𝑦
5 csbeq1a 3575 . . . . . . 7 (𝑥 = 𝑧𝐶 = 𝑧 / 𝑥𝐶)
65eqeq1d 2653 . . . . . 6 (𝑥 = 𝑧 → (𝐶 = 𝑦𝑧 / 𝑥𝐶 = 𝑦))
71, 2, 4, 6elrabf 3392 . . . . 5 (𝑧 ∈ {𝑥𝐵𝐶 = 𝑦} ↔ (𝑧𝐵𝑧 / 𝑥𝐶 = 𝑦))
8 ax-1 6 . . . . 5 (𝑧 / 𝑥𝐶 = 𝑦 → (𝑦𝐴𝑧 / 𝑥𝐶 = 𝑦))
97, 8simplbiim 659 . . . 4 (𝑧 ∈ {𝑥𝐵𝐶 = 𝑦} → (𝑦𝐴𝑧 / 𝑥𝐶 = 𝑦))
109impcom 445 . . 3 ((𝑦𝐴𝑧 ∈ {𝑥𝐵𝐶 = 𝑦}) → 𝑧 / 𝑥𝐶 = 𝑦)
1110rgen2 3004 . 2 𝑦𝐴𝑧 ∈ {𝑥𝐵𝐶 = 𝑦}𝑧 / 𝑥𝐶 = 𝑦
12 invdisj 4670 . 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 1523  wcel 2030  wral 2941  {crab 2945  csb 3566  Disj wdisj 4652
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
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-disj 4653
This theorem is referenced by:  disjxwrd  13501  disjwrdpfx  41733
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