Theorem wl-cbvalnaed 33319

Description: wl-cbvalnae 33320 with a context. (Contributed by Wolf Lammen, 28-Jul-2019.)

Hypotheses

Ref	Expression
wl-cbvalnaed.1	|- F/ x ph
wl-cbvalnaed.2	|- F/ y ph
wl-cbvalnaed.3	|- ( ph -> ( -. A. x x = y -> F/ y ps ) )
wl-cbvalnaed.4	|- ( ph -> ( -. A. x x = y -> F/ x ch ) )
wl-cbvalnaed.5	|- ( ph -> ( x = y -> ( ps <-> ch ) ) )

Assertion

Ref	Expression
wl-cbvalnaed	|- ( ph -> ( A. x ps <-> A. y ch ) )

Proof of Theorem wl-cbvalnaed

Step	Hyp	Ref	Expression
1		wl-cbvalnaed.1	. . . 4 |- F/ x ph
2		wl-cbvalnaed.2	. . . 4 |- F/ y ph
3		wl-cbvalnaed.5	. . . 4 |- ( ph -> ( x = y -> ( ps <-> ch ) ) )
4	1, 2, 3	wl-dral1d 33318	. . 3 |- ( ph -> ( A. x x = y -> ( A. x ps <-> A. y ch ) ) )
5	4	imp 445	. 2 |- ( ( ph /\ A. x x = y ) -> ( A. x ps <-> A. y ch ) )
6		nfnae 2318	. . . 4 |- F/ x -. A. x x = y
7	1, 6	nfan 1828	. . 3 |- F/ x ( ph /\ -. A. x x = y )
8		wl-nfnae1 33316	. . . 4 |- F/ y -. A. x x = y
9	2, 8	nfan 1828	. . 3 |- F/ y ( ph /\ -. A. x x = y )
10		wl-cbvalnaed.3	. . . 4 |- ( ph -> ( -. A. x x = y -> F/ y ps ) )
11	10	imp 445	. . 3 |- ( ( ph /\ -. A. x x = y ) -> F/ y ps )
12		wl-cbvalnaed.4	. . . 4 |- ( ph -> ( -. A. x x = y -> F/ x ch ) )
13	12	imp 445	. . 3 |- ( ( ph /\ -. A. x x = y ) -> F/ x ch )
14	3	adantr 481	. . 3 |- ( ( ph /\ -. A. x x = y ) -> ( x = y -> ( ps <-> ch ) ) )
15	7, 9, 11, 13, 14	cbv2 2270	. 2 |- ( ( ph /\ -. A. x x = y ) -> ( A. x ps <-> A. y ch ) )
16	5, 15	pm2.61dan 832	1 |- ( ph -> ( A. x ps <-> A. y ch ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   F/wnf 1708

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-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710

This theorem is referenced by:  wl-cbvalnae  33320