Theorem cbvdisjf 29385

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.

Step	Hyp	Ref	Expression
1		nfv 1843	. . . . . 6 |- F/ y x e. A
2		cbvdisjf.2	. . . . . . 7 |- F/_ y B
3	2	nfcri 2758	. . . . . 6 |- F/ y z e. B
4	1, 3	nfan 1828	. . . . 5 |- F/ y ( x e. A /\ z e. B )
5		cbvdisjf.1	. . . . . . 7 |- F/_ x A
6	5	nfcri 2758	. . . . . 6 |- F/ x y e. A
7		cbvdisjf.3	. . . . . . 7 |- F/_ x C
8	7	nfcri 2758	. . . . . 6 |- F/ x z e. C
9	6, 8	nfan 1828	. . . . 5 |- F/ x ( y e. A /\ z e. C )
10		eleq1 2689	. . . . . 6 |- ( x = y -> ( x e. A <-> y e. A ) )
11		cbvdisjf.4	. . . . . . 7 |- ( x = y -> B = C )
12	11	eleq2d 2687	. . . . . 6 |- ( x = y -> ( z e. B <-> z e. C ) )
13	10, 12	anbi12d 747	. . . . 5 |- ( x = y -> ( ( x e. A /\ z e. B ) <-> ( y e. A /\ z e. C ) ) )
14	4, 9, 13	cbvmo 2506	. . . 4 |- ( E* x ( x e. A /\ z e. B ) <-> E* y ( y e. A /\ z e. C ) )
15		df-rmo 2920	. . . 4 |- ( E* x e. A z e. B <-> E* x ( x e. A /\ z e. B ) )
16		df-rmo 2920	. . . 4 |- ( E* y e. A z e. C <-> E* y ( y e. A /\ z e. C ) )
17	14, 15, 16	3bitr4i 292	. . 3 |- ( E* x e. A z e. B <-> E* y e. A z e. C )
18	17	albii 1747	. 2 |- ( A. z E* x e. A z e. B <-> A. z E* y e. A z e. C )
19		df-disj 4621	. 2 |- (Disj x e. A B <-> A. z E* x e. A z e. B )
20		df-disj 4621	. 2 |- (Disj y e. A C <-> A. z E* y e. A z e. C )
21	18, 19, 20	3bitr4i 292	1 |- (Disj x e. A B <-> Disj y e. A C )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   E*wmo 2471   F/_wnfc 2751   E*wrmo 2915  Disj wdisj 4620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rmo 2920  df-disj 4621

This theorem is referenced by:  disjorsf  29393  ldgenpisyslem1  30226