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Theorem cbvdisjf 28172
Description: Change bound variables in a disjoint collection. (Contributed by Thierry Arnoux, 6-Apr-2017.)
Hypotheses
Ref Expression
cbvdisjf.1  |-  F/_ x A
cbvdisjf.2  |-  F/_ y B
cbvdisjf.3  |-  F/_ x C
cbvdisjf.4  |-  ( x  =  y  ->  B  =  C )
Assertion
Ref Expression
cbvdisjf  |-  (Disj  x  e.  A  B  <-> Disj  y  e.  A  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 1751 . . . . . 6  |-  F/ y  x  e.  A
2 cbvdisjf.2 . . . . . . 7  |-  F/_ y B
32nfcri 2577 . . . . . 6  |-  F/ y  z  e.  B
41, 3nfan 1984 . . . . 5  |-  F/ y ( x  e.  A  /\  z  e.  B
)
5 cbvdisjf.1 . . . . . . 7  |-  F/_ x A
65nfcri 2577 . . . . . 6  |-  F/ x  y  e.  A
7 cbvdisjf.3 . . . . . . 7  |-  F/_ x C
87nfcri 2577 . . . . . 6  |-  F/ x  z  e.  C
96, 8nfan 1984 . . . . 5  |-  F/ x
( y  e.  A  /\  z  e.  C
)
10 eleq1 2494 . . . . . 6  |-  ( x  =  y  ->  (
x  e.  A  <->  y  e.  A ) )
11 cbvdisjf.4 . . . . . . 7  |-  ( x  =  y  ->  B  =  C )
1211eleq2d 2492 . . . . . 6  |-  ( x  =  y  ->  (
z  e.  B  <->  z  e.  C ) )
1310, 12anbi12d 715 . . . . 5  |-  ( x  =  y  ->  (
( x  e.  A  /\  z  e.  B
)  <->  ( y  e.  A  /\  z  e.  C ) ) )
144, 9, 13cbvmo 2302 . . . 4  |-  ( E* x ( x  e.  A  /\  z  e.  B )  <->  E* y
( y  e.  A  /\  z  e.  C
) )
15 df-rmo 2783 . . . 4  |-  ( E* x  e.  A  z  e.  B  <->  E* x
( x  e.  A  /\  z  e.  B
) )
16 df-rmo 2783 . . . 4  |-  ( E* y  e.  A  z  e.  C  <->  E* y
( y  e.  A  /\  z  e.  C
) )
1714, 15, 163bitr4i 280 . . 3  |-  ( E* x  e.  A  z  e.  B  <->  E* y  e.  A  z  e.  C )
1817albii 1687 . 2  |-  ( A. z E* x  e.  A  z  e.  B  <->  A. z E* y  e.  A  z  e.  C )
19 df-disj 4392 . 2  |-  (Disj  x  e.  A  B  <->  A. z E* x  e.  A  z  e.  B )
20 df-disj 4392 . 2  |-  (Disj  y  e.  A  C  <->  A. z E* y  e.  A  z  e.  C )
2118, 19, 203bitr4i 280 1  |-  (Disj  x  e.  A  B  <-> Disj  y  e.  A  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370   A.wal 1435    = wceq 1437    e. wcel 1868   E*wmo 2266   F/_wnfc 2570   E*wrmo 2778  Disj wdisj 4391
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-cleq 2414  df-clel 2417  df-nfc 2572  df-rmo 2783  df-disj 4392
This theorem is referenced by:  disjorsf  28180  ldgenpisyslem1  28981
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