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Theorem cbvdisjf 24007
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 1629 . . . . . 6  |-  F/ y  x  e.  A
2 cbvdisjf.2 . . . . . . 7  |-  F/_ y B
32nfcri 2565 . . . . . 6  |-  F/ y  z  e.  B
41, 3nfan 1846 . . . . 5  |-  F/ y ( x  e.  A  /\  z  e.  B
)
5 cbvdisjf.1 . . . . . . 7  |-  F/_ x A
65nfcri 2565 . . . . . 6  |-  F/ x  y  e.  A
7 cbvdisjf.3 . . . . . . 7  |-  F/_ x C
87nfcri 2565 . . . . . 6  |-  F/ x  z  e.  C
96, 8nfan 1846 . . . . 5  |-  F/ x
( y  e.  A  /\  z  e.  C
)
10 eleq1 2495 . . . . . 6  |-  ( x  =  y  ->  (
x  e.  A  <->  y  e.  A ) )
11 cbvdisjf.4 . . . . . . 7  |-  ( x  =  y  ->  B  =  C )
1211eleq2d 2502 . . . . . 6  |-  ( x  =  y  ->  (
z  e.  B  <->  z  e.  C ) )
1310, 12anbi12d 692 . . . . 5  |-  ( x  =  y  ->  (
( x  e.  A  /\  z  e.  B
)  <->  ( y  e.  A  /\  z  e.  C ) ) )
144, 9, 13cbvmo 2317 . . . 4  |-  ( E* x ( x  e.  A  /\  z  e.  B )  <->  E* y
( y  e.  A  /\  z  e.  C
) )
15 df-rmo 2705 . . . 4  |-  ( E* x  e.  A z  e.  B  <->  E* x
( x  e.  A  /\  z  e.  B
) )
16 df-rmo 2705 . . . 4  |-  ( E* y  e.  A z  e.  C  <->  E* y
( y  e.  A  /\  z  e.  C
) )
1714, 15, 163bitr4i 269 . . 3  |-  ( E* x  e.  A z  e.  B  <->  E* y  e.  A z  e.  C
)
1817albii 1575 . 2  |-  ( A. z E* x  e.  A
z  e.  B  <->  A. z E* y  e.  A
z  e.  C )
19 df-disj 4175 . 2  |-  (Disj  x  e.  A B  <->  A. z E* x  e.  A
z  e.  B )
20 df-disj 4175 . 2  |-  (Disj  y  e.  A C  <->  A. z E* y  e.  A
z  e.  C )
2118, 19, 203bitr4i 269 1  |-  (Disj  x  e.  A B  <-> Disj  y  e.  A C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1549    = wceq 1652    e. wcel 1725   E*wmo 2281   F/_wnfc 2558   E*wrmo 2700  Disj wdisj 4174
This theorem is referenced by:  disjorsf  24014
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rmo 2705  df-disj 4175
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