Metamath Proof Explorer


Theorem nfun

Description: Bound-variable hypothesis builder for the union of classes. (Contributed by NM, 15-Sep-2003) (Revised by Mario Carneiro, 14-Oct-2016) Avoid ax-10 , ax-11 , ax-12 . (Revised by SN, 14-May-2025)

Ref Expression
Hypotheses nfun.1
|- F/_ x A
nfun.2
|- F/_ x B
Assertion nfun
|- F/_ x ( A u. B )

Proof

Step Hyp Ref Expression
1 nfun.1
 |-  F/_ x A
2 nfun.2
 |-  F/_ x B
3 elun
 |-  ( y e. ( A u. B ) <-> ( y e. A \/ y e. B ) )
4 1 nfcri
 |-  F/ x y e. A
5 2 nfcri
 |-  F/ x y e. B
6 4 5 nfor
 |-  F/ x ( y e. A \/ y e. B )
7 3 6 nfxfr
 |-  F/ x y e. ( A u. B )
8 7 nfci
 |-  F/_ x ( A u. B )