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Theorem invdisjrabw 5018
 Description: Version of invdisjrab 5019 with a disjoint variable condition, which does not require ax-13 2380. (Contributed by Gino Giotto, 26-Jan-2024.)
Assertion
Ref Expression
invdisjrabw 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 invdisjrabw
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfcv 2920 . . . . 5 𝑥𝑧
2 nfcv 2920 . . . . 5 𝑥𝐵
3 nfcsb1v 3830 . . . . . 6 𝑥𝑧 / 𝑥𝐶
43nfeq1 2935 . . . . 5 𝑥𝑧 / 𝑥𝐶 = 𝑦
5 csbeq1a 3820 . . . . . 6 (𝑥 = 𝑧𝐶 = 𝑧 / 𝑥𝐶)
65eqeq1d 2761 . . . . 5 (𝑥 = 𝑧 → (𝐶 = 𝑦𝑧 / 𝑥𝐶 = 𝑦))
71, 2, 4, 6elrabf 3599 . . . 4 (𝑧 ∈ {𝑥𝐵𝐶 = 𝑦} ↔ (𝑧𝐵𝑧 / 𝑥𝐶 = 𝑦))
8 simprr 773 . . . 4 ((𝑦𝐴 ∧ (𝑧𝐵𝑧 / 𝑥𝐶 = 𝑦)) → 𝑧 / 𝑥𝐶 = 𝑦)
97, 8sylan2b 597 . . 3 ((𝑦𝐴𝑧 ∈ {𝑥𝐵𝐶 = 𝑦}) → 𝑧 / 𝑥𝐶 = 𝑦)
109rgen2 3133 . 2 𝑦𝐴𝑧 ∈ {𝑥𝐵𝐶 = 𝑦}𝑧 / 𝑥𝐶 = 𝑦
11 invdisj 5017 . 2 (∀𝑦𝐴𝑧 ∈ {𝑥𝐵𝐶 = 𝑦}𝑧 / 𝑥𝐶 = 𝑦Disj 𝑦𝐴 {𝑥𝐵𝐶 = 𝑦})
1210, 11ax-mp 5 1 Disj 𝑦𝐴 {𝑥𝐵𝐶 = 𝑦}
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 400   = wceq 1539   ∈ wcel 2112  ∀wral 3071  {crab 3075  ⦋csb 3806  Disj wdisj 4998 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ral 3076  df-rmo 3079  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-disj 4999 This theorem is referenced by: (None)
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