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Metamath Proof Explorer

Theorem nfreud

Description: Deduction version of nfreu . Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 15-Feb-2013) (Revised by Mario Carneiro, 8-Oct-2016) (New usage is discouraged.)

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

Proof

Step Hyp Ref Expression
1 nfrmod.1 
 |-  F/ y ph
2 nfrmod.2 
 |-  ( ph -> F/_ x A )
3 nfrmod.3 
 |-  ( ph -> F/ x ps )
4 df-reu 
 |-  ( E! y e. A ps <-> E! 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 nfand 
 |-  ( ( ph /\ -. A. x x = y ) -> F/ x ( y e. A /\ ps ) )
11 1 10 nfeud2 
 |-  ( ph -> F/ x E! y ( y e. A /\ ps ) )
12 4 11 nfxfrd 
 |-  ( ph -> F/ x E! y e. A ps )