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Metamath Proof Explorer

Theorem bnj23

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (Proof shortened by Mario Carneiro, 22-Dec-2016) (New usage is discouraged.)

Ref Expression
Hypothesis bnj23.1 
|- B = { x e. A | -. ph }
Assertion bnj23 
|- ( A. z e. B -. z R y -> A. w e. A ( w R y -> [. w / x ]. ph ) )

Proof

Step Hyp Ref Expression
1 bnj23.1 
 |-  B = { x e. A | -. ph }
2 sbcng 
 |-  ( w e. _V -> ( [. w / x ]. -. ph <-> -. [. w / x ]. ph ) )
3 2 elv 
 |-  ( [. w / x ]. -. ph <-> -. [. w / x ]. ph )
4 1 eleq2i 
 |-  ( w e. B <-> w e. { x e. A | -. ph } )
5 nfcv 
 |-  F/_ x A
6 5 elrabsf 
 |-  ( w e. { x e. A | -. ph } <-> ( w e. A /\ [. w / x ]. -. ph ) )
7 4 6 bitri 
 |-  ( w e. B <-> ( w e. A /\ [. w / x ]. -. ph ) )
8 breq1 
 |-  ( z = w -> ( z R y <-> w R y ) )
9 8 notbid 
 |-  ( z = w -> ( -. z R y <-> -. w R y ) )
10 9 rspccv 
 |-  ( A. z e. B -. z R y -> ( w e. B -> -. w R y ) )
11 7 10 biimtrrid 
 |-  ( A. z e. B -. z R y -> ( ( w e. A /\ [. w / x ]. -. ph ) -> -. w R y ) )
12 11 expdimp 
 |-  ( ( A. z e. B -. z R y /\ w e. A ) -> ( [. w / x ]. -. ph -> -. w R y ) )
13 3 12 biimtrrid 
 |-  ( ( A. z e. B -. z R y /\ w e. A ) -> ( -. [. w / x ]. ph -> -. w R y ) )
14 13 con4d 
 |-  ( ( A. z e. B -. z R y /\ w e. A ) -> ( w R y -> [. w / x ]. ph ) )
15 14 ralrimiva 
 |-  ( A. z e. B -. z R y -> A. w e. A ( w R y -> [. w / x ]. ph ) )