Theorem rextpg 3778

Description: Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.)

Hypotheses

Ref	Expression
ralprg.1	|- (x = A -> (ph <-> ps))
ralprg.2	|- (x = B -> (ph <-> ch))
raltpg.3	|- (x = C -> (ph <-> th))

Assertion

Ref	Expression
rextpg	|- ((A e. V /\ B e. W /\ C e. X) -> (E.x e. {A, B, C}ph <-> (ps \/ ch \/ th)))

Distinct variable groups:   x,A   x,B   x,C   ps,x   ch,x   th,x

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

Proof of Theorem rextpg

Step	Hyp	Ref	Expression
1		ralprg.1	. . . . . 6 |- (x = A -> (ph <-> ps))
2		ralprg.2	. . . . . 6 |- (x = B -> (ph <-> ch))
3	1, 2	rexprg 3776	. . . . 5 |- ((A e. V /\ B e. W) -> (E.x e. {A, B}ph <-> (ps \/ ch)))
4	3	orbi1d 683	. . . 4 |- ((A e. V /\ B e. W) -> ((E.x e. {A, B}ph \/ E.x e. {C}ph) <-> ((ps \/ ch) \/ E.x e. {C}ph)))
5		raltpg.3	. . . . . 6 |- (x = C -> (ph <-> th))
6	5	rexsng 3766	. . . . 5 |- (C e. X -> (E.x e. {C}ph <-> th))
7	6	orbi2d 682	. . . 4 |- (C e. X -> (((ps \/ ch) \/ E.x e. {C}ph) <-> ((ps \/ ch) \/ th)))
8	4, 7	sylan9bb 680	. . 3 |- (((A e. V /\ B e. W) /\ C e. X) -> ((E.x e. {A, B}ph \/ E.x e. {C}ph) <-> ((ps \/ ch) \/ th)))
9	8	3impa 1146	. 2 |- ((A e. V /\ B e. W /\ C e. X) -> ((E.x e. {A, B}ph \/ E.x e. {C}ph) <-> ((ps \/ ch) \/ th)))
10		df-tp 3743	. . . 4 |- {A, B, C} = ({A, B} u. {C})
11	10	rexeqi 2812	. . 3 |- (E.x e. {A, B, C}ph <-> E.x e. ({A, B} u. {C})ph)
12		rexun 3443	. . 3 |- (E.x e. ({A, B} u. {C})ph <-> (E.x e. {A, B}ph \/ E.x e. {C}ph))
13	11, 12	bitri 240	. 2 |- (E.x e. {A, B, C}ph <-> (E.x e. {A, B}ph \/ E.x e. {C}ph))
14		df-3or 935	. 2 |- ((ps \/ ch \/ th) <-> ((ps \/ ch) \/ th))
15	9, 13, 14	3bitr4g 279	1 |- ((A e. V /\ B e. W /\ C e. X) -> (E.x e. {A, B, C}ph <-> (ps \/ ch \/ th)))

Syntax hints:   -> wi 4   <-> wb 176   \/ wo 357   /\ wa 358   \/ w3o 933   /\ w3a 934   = wceq 1642   e. wcel 1710  E.wrex 2615   u. cun 3207  {csn 3737  {cpr 3738  {ctp 3739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-un 3214  df-sn 3741  df-pr 3742  df-tp 3743

This theorem is referenced by:  rextp  3782