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Theorem cbvdisjf 29230
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 1840 . . . . . 6 𝑦 𝑥𝐴
2 cbvdisjf.2 . . . . . . 7 𝑦𝐵
32nfcri 2755 . . . . . 6 𝑦 𝑧𝐵
41, 3nfan 1825 . . . . 5 𝑦(𝑥𝐴𝑧𝐵)
5 cbvdisjf.1 . . . . . . 7 𝑥𝐴
65nfcri 2755 . . . . . 6 𝑥 𝑦𝐴
7 cbvdisjf.3 . . . . . . 7 𝑥𝐶
87nfcri 2755 . . . . . 6 𝑥 𝑧𝐶
96, 8nfan 1825 . . . . 5 𝑥(𝑦𝐴𝑧𝐶)
10 eleq1 2686 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
11 cbvdisjf.4 . . . . . . 7 (𝑥 = 𝑦𝐵 = 𝐶)
1211eleq2d 2684 . . . . . 6 (𝑥 = 𝑦 → (𝑧𝐵𝑧𝐶))
1310, 12anbi12d 746 . . . . 5 (𝑥 = 𝑦 → ((𝑥𝐴𝑧𝐵) ↔ (𝑦𝐴𝑧𝐶)))
144, 9, 13cbvmo 2505 . . . 4 (∃*𝑥(𝑥𝐴𝑧𝐵) ↔ ∃*𝑦(𝑦𝐴𝑧𝐶))
15 df-rmo 2915 . . . 4 (∃*𝑥𝐴 𝑧𝐵 ↔ ∃*𝑥(𝑥𝐴𝑧𝐵))
16 df-rmo 2915 . . . 4 (∃*𝑦𝐴 𝑧𝐶 ↔ ∃*𝑦(𝑦𝐴𝑧𝐶))
1714, 15, 163bitr4i 292 . . 3 (∃*𝑥𝐴 𝑧𝐵 ↔ ∃*𝑦𝐴 𝑧𝐶)
1817albii 1744 . 2 (∀𝑧∃*𝑥𝐴 𝑧𝐵 ↔ ∀𝑧∃*𝑦𝐴 𝑧𝐶)
19 df-disj 4584 . 2 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑧∃*𝑥𝐴 𝑧𝐵)
20 df-disj 4584 . 2 (Disj 𝑦𝐴 𝐶 ↔ ∀𝑧∃*𝑦𝐴 𝑧𝐶)
2118, 19, 203bitr4i 292 1 (Disj 𝑥𝐴 𝐵Disj 𝑦𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1478   = wceq 1480  wcel 1987  ∃*wmo 2470  wnfc 2748  ∃*wrmo 2910  Disj wdisj 4583
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-cleq 2614  df-clel 2617  df-nfc 2750  df-rmo 2915  df-disj 4584
This theorem is referenced by:  disjorsf  29238  ldgenpisyslem1  30007
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