Metamath Proof Explorer


Theorem cbviunv

Description: Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 15-Sep-2003) Add disjoint variable condition to avoid ax-13 . See cbviunvg for a less restrictive version requiring more axioms. (Revised by GG, 14-Aug-2025)

Ref Expression
Hypothesis cbviunv.1 ( 𝑥 = 𝑦𝐵 = 𝐶 )
Assertion cbviunv 𝑥𝐴 𝐵 = 𝑦𝐴 𝐶

Proof

Step Hyp Ref Expression
1 cbviunv.1 ( 𝑥 = 𝑦𝐵 = 𝐶 )
2 1 eleq2d ( 𝑥 = 𝑦 → ( 𝑧𝐵𝑧𝐶 ) )
3 2 cbvrexvw ( ∃ 𝑥𝐴 𝑧𝐵 ↔ ∃ 𝑦𝐴 𝑧𝐶 )
4 3 abbii { 𝑧 ∣ ∃ 𝑥𝐴 𝑧𝐵 } = { 𝑧 ∣ ∃ 𝑦𝐴 𝑧𝐶 }
5 df-iun 𝑥𝐴 𝐵 = { 𝑧 ∣ ∃ 𝑥𝐴 𝑧𝐵 }
6 df-iun 𝑦𝐴 𝐶 = { 𝑧 ∣ ∃ 𝑦𝐴 𝑧𝐶 }
7 4 5 6 3eqtr4i 𝑥𝐴 𝐵 = 𝑦𝐴 𝐶