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Metamath Proof Explorer

Theorem nfrald

Description: Deduction version of nfral . Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker nfraldw when possible. (Contributed by NM, 15-Feb-2013) (Revised by Mario Carneiro, 7-Oct-2016) (New usage is discouraged.)

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

Proof

Step Hyp Ref Expression
1 nfrald.1 
 |-  F/ y ph
2 nfrald.2 
 |-  ( ph -> F/_ x A )
3 nfrald.3 
 |-  ( ph -> F/ x ps )
4 df-ral 
 |-  ( A. y e. A ps <-> A. y ( y e. A -> ps ) )
5 nfcvf 
 |-  ( -. A. x x = y -> F/_ x y )
6 5 adantl 
 |-  ( ( ph /\ -. A. x x = y ) -> F/_ x y )
7 2 adantr 
 |-  ( ( ph /\ -. A. x x = y ) -> F/_ x A )
8 6 7 nfeld 
 |-  ( ( ph /\ -. A. x x = y ) -> F/ x y e. A )
9 3 adantr 
 |-  ( ( ph /\ -. A. x x = y ) -> F/ x ps )
10 8 9 nfimd 
 |-  ( ( ph /\ -. A. x x = y ) -> F/ x ( y e. A -> ps ) )
11 1 10 nfald2 
 |-  ( ph -> F/ x A. y ( y e. A -> ps ) )
12 4 11 nfxfrd 
 |-  ( ph -> F/ x A. y e. A ps )