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Theorem cbvdisjf 31375
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 1917 . . . . . 6 𝑦 𝑥𝐴
2 cbvdisjf.2 . . . . . . 7 𝑦𝐵
32nfcri 2892 . . . . . 6 𝑦 𝑧𝐵
41, 3nfan 1902 . . . . 5 𝑦(𝑥𝐴𝑧𝐵)
5 cbvdisjf.1 . . . . . . 7 𝑥𝐴
65nfcri 2892 . . . . . 6 𝑥 𝑦𝐴
7 cbvdisjf.3 . . . . . . 7 𝑥𝐶
87nfcri 2892 . . . . . 6 𝑥 𝑧𝐶
96, 8nfan 1902 . . . . 5 𝑥(𝑦𝐴𝑧𝐶)
10 eleq1w 2820 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
11 cbvdisjf.4 . . . . . . 7 (𝑥 = 𝑦𝐵 = 𝐶)
1211eleq2d 2823 . . . . . 6 (𝑥 = 𝑦 → (𝑧𝐵𝑧𝐶))
1310, 12anbi12d 631 . . . . 5 (𝑥 = 𝑦 → ((𝑥𝐴𝑧𝐵) ↔ (𝑦𝐴𝑧𝐶)))
144, 9, 13cbvmow 2601 . . . 4 (∃*𝑥(𝑥𝐴𝑧𝐵) ↔ ∃*𝑦(𝑦𝐴𝑧𝐶))
15 df-rmo 3351 . . . 4 (∃*𝑥𝐴 𝑧𝐵 ↔ ∃*𝑥(𝑥𝐴𝑧𝐵))
16 df-rmo 3351 . . . 4 (∃*𝑦𝐴 𝑧𝐶 ↔ ∃*𝑦(𝑦𝐴𝑧𝐶))
1714, 15, 163bitr4i 302 . . 3 (∃*𝑥𝐴 𝑧𝐵 ↔ ∃*𝑦𝐴 𝑧𝐶)
1817albii 1821 . 2 (∀𝑧∃*𝑥𝐴 𝑧𝐵 ↔ ∀𝑧∃*𝑦𝐴 𝑧𝐶)
19 df-disj 5069 . 2 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑧∃*𝑥𝐴 𝑧𝐵)
20 df-disj 5069 . 2 (Disj 𝑦𝐴 𝐶 ↔ ∀𝑧∃*𝑦𝐴 𝑧𝐶)
2118, 19, 203bitr4i 302 1 (Disj 𝑥𝐴 𝐵Disj 𝑦𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wal 1539   = wceq 1541  wcel 2106  ∃*wmo 2536  wnfc 2885  ∃*wrmo 3350  Disj wdisj 5068
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-12 2171  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-mo 2538  df-cleq 2728  df-clel 2814  df-nfc 2887  df-rmo 3351  df-disj 5069
This theorem is referenced by:  disjorsf  31384  ldgenpisyslem1  32631
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