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Theorem cbvdisjf 23365
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 1609 . . . . . 6  |-  F/ y  x  e.  A
2 cbvdisjf.2 . . . . . . 7  |-  F/_ y B
32nfcri 2426 . . . . . 6  |-  F/ y  z  e.  B
41, 3nfan 1783 . . . . 5  |-  F/ y ( x  e.  A  /\  z  e.  B
)
5 nfcv 2432 . . . . . . 7  |-  F/_ x
y
6 cbvdisjf.1 . . . . . . 7  |-  F/_ x A
75, 6nfel 2440 . . . . . 6  |-  F/ x  y  e.  A
8 cbvdisjf.3 . . . . . . 7  |-  F/_ x C
98nfcri 2426 . . . . . 6  |-  F/ x  z  e.  C
107, 9nfan 1783 . . . . 5  |-  F/ x
( y  e.  A  /\  z  e.  C
)
11 eleq1 2356 . . . . . 6  |-  ( x  =  y  ->  (
x  e.  A  <->  y  e.  A ) )
12 cbvdisjf.4 . . . . . . 7  |-  ( x  =  y  ->  B  =  C )
1312eleq2d 2363 . . . . . 6  |-  ( x  =  y  ->  (
z  e.  B  <->  z  e.  C ) )
1411, 13anbi12d 691 . . . . 5  |-  ( x  =  y  ->  (
( x  e.  A  /\  z  e.  B
)  <->  ( y  e.  A  /\  z  e.  C ) ) )
154, 10, 14cbvmo 2193 . . . 4  |-  ( E* x ( x  e.  A  /\  z  e.  B )  <->  E* y
( y  e.  A  /\  z  e.  C
) )
16 df-rmo 2564 . . . 4  |-  ( E* x  e.  A z  e.  B  <->  E* x
( x  e.  A  /\  z  e.  B
) )
17 df-rmo 2564 . . . 4  |-  ( E* y  e.  A z  e.  C  <->  E* y
( y  e.  A  /\  z  e.  C
) )
1815, 16, 173bitr4i 268 . . 3  |-  ( E* x  e.  A z  e.  B  <->  E* y  e.  A z  e.  C
)
1918albii 1556 . 2  |-  ( A. z E* x  e.  A
z  e.  B  <->  A. z E* y  e.  A
z  e.  C )
20 df-disj 4010 . 2  |-  (Disj  x  e.  A B  <->  A. z E* x  e.  A
z  e.  B )
21 df-disj 4010 . 2  |-  (Disj  y  e.  A C  <->  A. z E* y  e.  A
z  e.  C )
2219, 20, 213bitr4i 268 1  |-  (Disj  x  e.  A B  <-> Disj  y  e.  A C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530    = wceq 1632    e. wcel 1696   E*wmo 2157   F/_wnfc 2419   E*wrmo 2559  Disj wdisj 4009
This theorem is referenced by:  disjorsf  23372
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rmo 2564  df-disj 4010
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