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Theorem invdisjrab 4612
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 2761 . . . . . 6 𝑥𝑧
2 nfcv 2761 . . . . . 6 𝑥𝐵
3 nfcsb1v 3535 . . . . . . 7 𝑥𝑧 / 𝑥𝐶
43nfeq1 2774 . . . . . 6 𝑥𝑧 / 𝑥𝐶 = 𝑦
5 csbeq1a 3528 . . . . . . 7 (𝑥 = 𝑧𝐶 = 𝑧 / 𝑥𝐶)
65eqeq1d 2623 . . . . . 6 (𝑥 = 𝑧 → (𝐶 = 𝑦𝑧 / 𝑥𝐶 = 𝑦))
71, 2, 4, 6elrabf 3348 . . . . 5 (𝑧 ∈ {𝑥𝐵𝐶 = 𝑦} ↔ (𝑧𝐵𝑧 / 𝑥𝐶 = 𝑦))
8 ax-1 6 . . . . 5 (𝑧 / 𝑥𝐶 = 𝑦 → (𝑦𝐴𝑧 / 𝑥𝐶 = 𝑦))
97, 8simplbiim 658 . . . 4 (𝑧 ∈ {𝑥𝐵𝐶 = 𝑦} → (𝑦𝐴𝑧 / 𝑥𝐶 = 𝑦))
109impcom 446 . . 3 ((𝑦𝐴𝑧 ∈ {𝑥𝐵𝐶 = 𝑦}) → 𝑧 / 𝑥𝐶 = 𝑦)
1110rgen2 2971 . 2 𝑦𝐴𝑧 ∈ {𝑥𝐵𝐶 = 𝑦}𝑧 / 𝑥𝐶 = 𝑦
12 invdisj 4611 . 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 1480  wcel 1987  wral 2908  {crab 2912  csb 3519  Disj wdisj 4593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-disj 4594
This theorem is referenced by:  disjxwrd  13409  disjwrdpfx  40737
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