Metamath Proof Explorer


Theorem nfiung

Description: Bound-variable hypothesis builder for indexed union. Usage of this theorem is discouraged because it depends on ax-13 . See nfiun for a version with more disjoint variable conditions, but not requiring ax-13 . (Contributed by Mario Carneiro, 25-Jan-2014) (New usage is discouraged.)

Ref Expression
Hypotheses nfiung.1 𝑦 𝐴
nfiung.2 𝑦 𝐵
Assertion nfiung 𝑦 𝑥𝐴 𝐵

Proof

Step Hyp Ref Expression
1 nfiung.1 𝑦 𝐴
2 nfiung.2 𝑦 𝐵
3 df-iun 𝑥𝐴 𝐵 = { 𝑧 ∣ ∃ 𝑥𝐴 𝑧𝐵 }
4 2 nfcri 𝑦 𝑧𝐵
5 1 4 nfrexg 𝑦𝑥𝐴 𝑧𝐵
6 5 nfabg 𝑦 { 𝑧 ∣ ∃ 𝑥𝐴 𝑧𝐵 }
7 3 6 nfcxfr 𝑦 𝑥𝐴 𝐵