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Theorem cbvdisjf 30337
Description: Change bound variables in a disjoint collection. (Contributed by Thierry Arnoux, 6-Apr-2017.)
Hypotheses
Ref Expression
cbvdisjf.1 𝑥𝐴
cbvdisjf.2 𝑦𝐵
cbvdisjf.3 𝑥𝐶
cbvdisjf.4 (𝑥 = 𝑦𝐵 = 𝐶)
Assertion
Ref Expression
cbvdisjf (Disj 𝑥𝐴 𝐵Disj 𝑦𝐴 𝐶)
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem cbvdisjf
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfv 1915 . . . . . 6 𝑦 𝑥𝐴
2 cbvdisjf.2 . . . . . . 7 𝑦𝐵
32nfcri 2946 . . . . . 6 𝑦 𝑧𝐵
41, 3nfan 1900 . . . . 5 𝑦(𝑥𝐴𝑧𝐵)
5 cbvdisjf.1 . . . . . . 7 𝑥𝐴
65nfcri 2946 . . . . . 6 𝑥 𝑦𝐴
7 cbvdisjf.3 . . . . . . 7 𝑥𝐶
87nfcri 2946 . . . . . 6 𝑥 𝑧𝐶
96, 8nfan 1900 . . . . 5 𝑥(𝑦𝐴𝑧𝐶)
10 eleq1w 2875 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
11 cbvdisjf.4 . . . . . . 7 (𝑥 = 𝑦𝐵 = 𝐶)
1211eleq2d 2878 . . . . . 6 (𝑥 = 𝑦 → (𝑧𝐵𝑧𝐶))
1310, 12anbi12d 633 . . . . 5 (𝑥 = 𝑦 → ((𝑥𝐴𝑧𝐵) ↔ (𝑦𝐴𝑧𝐶)))
144, 9, 13cbvmow 2666 . . . 4 (∃*𝑥(𝑥𝐴𝑧𝐵) ↔ ∃*𝑦(𝑦𝐴𝑧𝐶))
15 df-rmo 3117 . . . 4 (∃*𝑥𝐴 𝑧𝐵 ↔ ∃*𝑥(𝑥𝐴𝑧𝐵))
16 df-rmo 3117 . . . 4 (∃*𝑦𝐴 𝑧𝐶 ↔ ∃*𝑦(𝑦𝐴𝑧𝐶))
1714, 15, 163bitr4i 306 . . 3 (∃*𝑥𝐴 𝑧𝐵 ↔ ∃*𝑦𝐴 𝑧𝐶)
1817albii 1821 . 2 (∀𝑧∃*𝑥𝐴 𝑧𝐵 ↔ ∀𝑧∃*𝑦𝐴 𝑧𝐶)
19 df-disj 4999 . 2 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑧∃*𝑥𝐴 𝑧𝐵)
20 df-disj 4999 . 2 (Disj 𝑦𝐴 𝐶 ↔ ∀𝑧∃*𝑦𝐴 𝑧𝐶)
2118, 19, 203bitr4i 306 1 (Disj 𝑥𝐴 𝐵Disj 𝑦𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1536   = wceq 1538  wcel 2112  ∃*wmo 2599  wnfc 2939  ∃*wrmo 3112  Disj wdisj 4998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-mo 2601  df-cleq 2794  df-clel 2873  df-nfc 2941  df-rmo 3117  df-disj 4999
This theorem is referenced by:  disjorsf  30346  ldgenpisyslem1  31530
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