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Theorem raltpd 4092
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
StepHypRef Expression
1 an3andi 1341 . . . . . 6  |-  ( (
ph  /\  ( ch  /\ 
th  /\  ta )
)  <->  ( ( ph  /\ 
ch )  /\  ( ph  /\  th )  /\  ( ph  /\  ta )
) )
21a1i 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 ) )
76expcom 433 . . . . . . . 8  |-  ( x  =  A  ->  ( ph  ->  ( ps  <->  ch )
) )
87pm5.32d 637 . . . . . . 7  |-  ( x  =  A  ->  (
( ph  /\  ps )  <->  (
ph  /\  ch )
) )
9 ralprd.2 . . . . . . . . 9  |-  ( (
ph  /\  x  =  B )  ->  ( ps 
<->  th ) )
109expcom 433 . . . . . . . 8  |-  ( x  =  B  ->  ( ph  ->  ( ps  <->  th )
) )
1110pm5.32d 637 . . . . . . 7  |-  ( x  =  B  ->  (
( ph  /\  ps )  <->  (
ph  /\  th )
) )
12 raltpd.3 . . . . . . . . 9  |-  ( (
ph  /\  x  =  C )  ->  ( ps 
<->  ta ) )
1312expcom 433 . . . . . . . 8  |-  ( x  =  C  ->  ( ph  ->  ( ps  <->  ta )
) )
1413pm5.32d 637 . . . . . . 7  |-  ( x  =  C  ->  (
( ph  /\  ps )  <->  (
ph  /\  ta )
) )
158, 11, 14raltpg 4020 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( A. x  e. 
{ A ,  B ,  C }  ( ph  /\ 
ps )  <->  ( ( ph  /\  ch )  /\  ( ph  /\  th )  /\  ( ph  /\  ta ) ) ) )
163, 4, 5, 15syl3anc 1228 . . . . 5  |-  ( ph  ->  ( A. x  e. 
{ A ,  B ,  C }  ( ph  /\ 
ps )  <->  ( ( ph  /\  ch )  /\  ( ph  /\  th )  /\  ( ph  /\  ta ) ) ) )
173tpnzd 4091 . . . . . 6  |-  ( ph  ->  { A ,  B ,  C }  =/=  (/) )
18 r19.28zv 3865 . . . . . 6  |-  ( { A ,  B ,  C }  =/=  (/)  ->  ( A. x  e.  { A ,  B ,  C } 
( ph  /\  ps )  <->  (
ph  /\  A. x  e.  { A ,  B ,  C } ps )
) )
1917, 18syl 17 . . . . 5  |-  ( ph  ->  ( A. x  e. 
{ A ,  B ,  C }  ( ph  /\ 
ps )  <->  ( ph  /\ 
A. x  e.  { A ,  B ,  C } ps ) ) )
202, 16, 193bitr2d 281 . . . 4  |-  ( ph  ->  ( ( ph  /\  ( ch  /\  th  /\  ta ) )  <->  ( ph  /\ 
A. x  e.  { A ,  B ,  C } ps ) ) )
2120bianabs 879 . . 3  |-  ( ph  ->  ( ( ph  /\  ( ch  /\  th  /\  ta ) )  <->  A. x  e.  { A ,  B ,  C } ps )
)
2221bicomd 201 . 2  |-  ( ph  ->  ( A. x  e. 
{ A ,  B ,  C } ps  <->  ( ph  /\  ( ch  /\  th  /\  ta ) ) ) )
2322bianabs 879 1  |-  ( ph  ->  ( A. x  e. 
{ A ,  B ,  C } ps  <->  ( ch  /\ 
th  /\  ta )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 972    = wceq 1403    e. wcel 1840    =/= wne 2596   A.wral 2751   (/)c0 3735   {ctp 3973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-v 3058  df-sbc 3275  df-dif 3414  df-un 3416  df-nul 3736  df-sn 3970  df-pr 3972  df-tp 3974
This theorem is referenced by:  trgcgrg  24182
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