Theorem nfdisj1 4633

Description: Bound-variable hypothesis builder for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)

Assertion

Ref	Expression
nfdisj1	|- F/ xDisj x e. A B

Proof of Theorem nfdisj1

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-disj 4621	. 2 |- (Disj x e. A B <-> A. y E* x e. A y e. B )
2		nfrmo1 3111	. . 3 |- F/ x E* x e. A y e. B
3	2	nfal 2153	. 2 |- F/ x A. y E* x e. A y e. B
4	1, 3	nfxfr 1779	1 |- F/ xDisj x e. A B

Syntax hints:   A.wal 1481   F/wnf 1708   e. wcel 1990   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-10 2019  ax-11 2034  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-eu 2474  df-mo 2475  df-rmo 2920  df-disj 4621

This theorem is referenced by:  disjabrex  29395  disjabrexf  29396  hasheuni  30147  ldgenpisyslem1  30226  measvunilem  30275  measvunilem0  30276  measvuni  30277  measinblem  30283  voliune  30292  volfiniune  30293  volmeas  30294  dstrvprob  30533  ismeannd  40684