Metamath Proof Explorer


Theorem nfiund

Description: Bound-variable hypothesis builder for indexed union. (Contributed by Emmett Weisz, 6-Dec-2019) Add disjoint variable condition to avoid ax-13 . See nfiundg for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024)

Ref Expression
Hypotheses nfiund.1 xφ
nfiund.2 φ_yA
nfiund.3 φ_yB
Assertion nfiund φ_yxAB

Proof

Step Hyp Ref Expression
1 nfiund.1 xφ
2 nfiund.2 φ_yA
3 nfiund.3 φ_yB
4 df-iun xAB=z|xAzB
5 nfv zφ
6 3 nfcrd φyzB
7 1 2 6 nfrexdw φyxAzB
8 5 7 nfabdw φ_yz|xAzB
9 4 8 nfcxfrd φ_yxAB