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Metamath Proof Explorer

Theorem nfraldw

Description: Deduction version of nfralw . Version of nfrald with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 15-Feb-2013) Avoid ax-9 , ax-ext . (Revised by GG, 24-Sep-2024)

Ref Expression
Hypotheses nfraldw.1 
|- F/ y ph
nfraldw.2 
|- ( ph -> F/_ x A )
nfraldw.3 
|- ( ph -> F/ x ps )
Assertion nfraldw 
|- ( ph -> F/ x A. y e. A ps )

Proof

Step Hyp Ref Expression
1 nfraldw.1 
 |-  F/ y ph
2 nfraldw.2 
 |-  ( ph -> F/_ x A )
3 nfraldw.3 
 |-  ( ph -> F/ x ps )
4 df-ral 
 |-  ( A. y e. A ps <-> A. y ( y e. A -> ps ) )
5 2 nfcrd 
 |-  ( ph -> F/ x y e. A )
6 5 3 nfimd 
 |-  ( ph -> F/ x ( y e. A -> ps ) )
7 1 6 nfald 
 |-  ( ph -> F/ x A. y ( y e. A -> ps ) )
8 4 7 nfxfrd 
 |-  ( ph -> F/ x A. y e. A ps )