Metamath Proof Explorer


Theorem cbvrabw

Description: Rule to change the bound variable in a restricted class abstraction, using implicit substitution. Version of cbvrab with a disjoint variable condition, which does not require ax-13 . (Contributed by Andrew Salmon, 11-Jul-2011) Avoid ax-13 . (Revised by GG, 10-Jan-2024) Avoid ax-10 . (Revised by Wolf Lammen, 19-Jul-2025)

Ref Expression
Hypotheses cbvrabw.1 _ x A
cbvrabw.2 _ y A
cbvrabw.3 y φ
cbvrabw.4 x ψ
cbvrabw.5 x = y φ ψ
Assertion cbvrabw x A | φ = y A | ψ

Proof

Step Hyp Ref Expression
1 cbvrabw.1 _ x A
2 cbvrabw.2 _ y A
3 cbvrabw.3 y φ
4 cbvrabw.4 x ψ
5 cbvrabw.5 x = y φ ψ
6 2 nfcri y x A
7 6 3 nfan y x A φ
8 1 nfcri x y A
9 8 4 nfan x y A ψ
10 eleq1w x = y x A y A
11 10 5 anbi12d x = y x A φ y A ψ
12 7 9 11 cbvabw x | x A φ = y | y A ψ
13 df-rab x A | φ = x | x A φ
14 df-rab y A | ψ = y | y A ψ
15 12 13 14 3eqtr4i x A | φ = y A | ψ