Theorem raltpd 4315

Description: Convert a quantification over a triple to a conjunction. (Contributed by Thierry Arnoux, 8-Apr-2019.)

Hypotheses

Ref	Expression
ralprd.1	|- ( ( ph /\ x = A ) -> ( ps <-> ch ) )
ralprd.2	|- ( ( ph /\ x = B ) -> ( ps <-> th ) )
raltpd.3	|- ( ( ph /\ x = C ) -> ( ps <-> ta ) )
ralprd.a	|- ( ph -> A e. V )
ralprd.b	|- ( ph -> B e. W )
raltpd.c	|- ( ph -> C e. X )

Assertion

Ref	Expression
raltpd	|- ( ph -> ( A. x e. { A , B , C } ps <-> ( ch /\ th /\ ta ) ) )

Distinct variable groups:   x, A   x, B   x, C   ph, x   ch, x   th, x   ta, x

Allowed substitution hints:   ps( x)   V( x)   W( x)   X( x)

Proof of Theorem raltpd

Step	Hyp	Ref	Expression
1		an3andi 1445	. . . . . 6 |- ( ( ph /\ ( ch /\ th /\ ta ) ) <-> ( ( ph /\ ch ) /\ ( ph /\ th ) /\ ( ph /\ ta ) ) )
2	1	a1i 11	. . . . 5 |- ( ph -> ( ( ph /\ ( ch /\ th /\ ta ) ) <-> ( ( ph /\ ch ) /\ ( ph /\ th ) /\ ( ph /\ ta ) ) ) )
3		ralprd.a	. . . . . 6 |- ( ph -> A e. V )
4		ralprd.b	. . . . . 6 |- ( ph -> B e. W )
5		raltpd.c	. . . . . 6 |- ( ph -> C e. X )
6		ralprd.1	. . . . . . . . 9 |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )
7	6	expcom 451	. . . . . . . 8 |- ( x = A -> ( ph -> ( ps <-> ch ) ) )
8	7	pm5.32d 671	. . . . . . 7 |- ( x = A -> ( ( ph /\ ps ) <-> ( ph /\ ch ) ) )
9		ralprd.2	. . . . . . . . 9 |- ( ( ph /\ x = B ) -> ( ps <-> th ) )
10	9	expcom 451	. . . . . . . 8 |- ( x = B -> ( ph -> ( ps <-> th ) ) )
11	10	pm5.32d 671	. . . . . . 7 |- ( x = B -> ( ( ph /\ ps ) <-> ( ph /\ th ) ) )
12		raltpd.3	. . . . . . . . 9 |- ( ( ph /\ x = C ) -> ( ps <-> ta ) )
13	12	expcom 451	. . . . . . . 8 |- ( x = C -> ( ph -> ( ps <-> ta ) ) )
14	13	pm5.32d 671	. . . . . . 7 |- ( x = C -> ( ( ph /\ ps ) <-> ( ph /\ ta ) ) )
15	8, 11, 14	raltpg 4236	. . . . . 6 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( A. x e. { A , B , C } ( ph /\ ps ) <-> ( ( ph /\ ch ) /\ ( ph /\ th ) /\ ( ph /\ ta ) ) ) )
16	3, 4, 5, 15	syl3anc 1326	. . . . 5 |- ( ph -> ( A. x e. { A , B , C } ( ph /\ ps ) <-> ( ( ph /\ ch ) /\ ( ph /\ th ) /\ ( ph /\ ta ) ) ) )
17	3	tpnzd 4314	. . . . . 6 |- ( ph -> { A , B , C } =/= (/) )
18		r19.28zv 4066	. . . . . 6 |- ( { A , B , C } =/= (/) -> ( A. x e. { A , B , C } ( ph /\ ps ) <-> ( ph /\ A. x e. { A , B , C } ps ) ) )
19	17, 18	syl 17	. . . . 5 |- ( ph -> ( A. x e. { A , B , C } ( ph /\ ps ) <-> ( ph /\ A. x e. { A , B , C } ps ) ) )
20	2, 16, 19	3bitr2d 296	. . . 4 |- ( ph -> ( ( ph /\ ( ch /\ th /\ ta ) ) <-> ( ph /\ A. x e. { A , B , C } ps ) ) )
21	20	bianabs 924	. . 3 |- ( ph -> ( ( ph /\ ( ch /\ th /\ ta ) ) <-> A. x e. { A , B , C } ps ) )
22	21	bicomd 213	. 2 |- ( ph -> ( A. x e. { A , B , C } ps <-> ( ph /\ ( ch /\ th /\ ta ) ) ) )
23	22	bianabs 924	1 |- ( ph -> ( A. x e. { A , B , C } ps <-> ( ch /\ th /\ ta ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   (/)c0 3915   {ctp 4181

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-3or 1038  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-ne 2795  df-ral 2917  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-nul 3916  df-sn 4178  df-pr 4180  df-tp 4182

This theorem is referenced by:  eqwrds3  13704  trgcgrg  25410  tgcgr4  25426  cplgr3v  26331