Theorem bnj23 30784

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.)

Hypothesis

Ref	Expression
bnj23.1	|- B = { x e. A | -. ph }

Assertion

Ref	Expression
bnj23	|- ( A. z e. B -. z R y -> A. w e. A ( w R y -> [. w / x ]. ph ) )

Distinct variable groups:   x, A   y, A, z   w, B, y, z   w, R, y, z

Allowed substitution hints:   ph( x, y, z, w)   A( w)   B( x)   R( x)

Proof of Theorem bnj23

Step	Hyp	Ref	Expression
1		vex 3203	. . . . 5 |- w e. _V
2		sbcng 3476	. . . . 5 |- ( w e. _V -> ( [. w / x ]. -. ph <-> -. [. w / x ]. ph ) )
3	1, 2	ax-mp 5	. . . 4 |- ( [. w / x ]. -. ph <-> -. [. w / x ]. ph )
4		bnj23.1	. . . . . . . 8 |- B = { x e. A | -. ph }
5	4	eleq2i 2693	. . . . . . 7 |- ( w e. B <-> w e. { x e. A | -. ph } )
6		nfcv 2764	. . . . . . . 8 |- F/_ x A
7	6	elrabsf 3474	. . . . . . 7 |- ( w e. { x e. A | -. ph } <-> ( w e. A /\ [. w / x ]. -. ph ) )
8	5, 7	bitri 264	. . . . . 6 |- ( w e. B <-> ( w e. A /\ [. w / x ]. -. ph ) )
9		breq1 4656	. . . . . . . 8 |- ( z = w -> ( z R y <-> w R y ) )
10	9	notbid 308	. . . . . . 7 |- ( z = w -> ( -. z R y <-> -. w R y ) )
11	10	rspccv 3306	. . . . . 6 |- ( A. z e. B -. z R y -> ( w e. B -> -. w R y ) )
12	8, 11	syl5bir 233	. . . . 5 |- ( A. z e. B -. z R y -> ( ( w e. A /\ [. w / x ]. -. ph ) -> -. w R y ) )
13	12	expdimp 453	. . . 4 |- ( ( A. z e. B -. z R y /\ w e. A ) -> ( [. w / x ]. -. ph -> -. w R y ) )
14	3, 13	syl5bir 233	. . 3 |- ( ( A. z e. B -. z R y /\ w e. A ) -> ( -. [. w / x ]. ph -> -. w R y ) )
15	14	con4d 114	. 2 |- ( ( A. z e. B -. z R y /\ w e. A ) -> ( w R y -> [. w / x ]. ph ) )
16	15	ralrimiva 2966	1 |- ( A. z e. B -. z R y -> A. w e. A ( w R y -> [. w / x ]. ph ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   [.wsbc 3435   class class class wbr 4653

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654

This theorem is referenced by:  bnj110  30928