Theorem raltpg 3778

Description: Convert a quantification over a triple to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised 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
raltpg	|- ((A e. V /\ B e. W /\ C e. X) -> (A.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 raltpg

Step	Hyp	Ref	Expression
1		ralprg.1	. . . . 5 |- (x = A -> (ph <-> ps))
2		ralprg.2	. . . . 5 |- (x = B -> (ph <-> ch))
3	1, 2	ralprg 3776	. . . 4 |- ((A e. V /\ B e. W) -> (A.x e. {A, B}ph <-> (ps /\ ch)))
4		raltpg.3	. . . . 5 |- (x = C -> (ph <-> th))
5	4	ralsng 3766	. . . 4 |- (C e. X -> (A.x e. {C}ph <-> th))
6	3, 5	bi2anan9 843	. . 3 |- (((A e. V /\ B e. W) /\ C e. X) -> ((A.x e. {A, B}ph /\ A.x e. {C}ph) <-> ((ps /\ ch) /\ th)))
7	6	3impa 1146	. 2 |- ((A e. V /\ B e. W /\ C e. X) -> ((A.x e. {A, B}ph /\ A.x e. {C}ph) <-> ((ps /\ ch) /\ th)))
8		df-tp 3744	. . . 4 |- {A, B, C} = ({A, B} u. {C})
9	8	raleqi 2812	. . 3 |- (A.x e. {A, B, C}ph <-> A.x e. ({A, B} u. {C})ph)
10		ralunb 3445	. . 3 |- (A.x e. ({A, B} u. {C})ph <-> (A.x e. {A, B}ph /\ A.x e. {C}ph))
11	9, 10	bitri 240	. 2 |- (A.x e. {A, B, C}ph <-> (A.x e. {A, B}ph /\ A.x e. {C}ph))
12		df-3an 936	. 2 |- ((ps /\ ch /\ th) <-> ((ps /\ ch) /\ th))
13	7, 11, 12	3bitr4g 279	1 |- ((A e. V /\ B e. W /\ C e. X) -> (A.x e. {A, B, C}ph <-> (ps /\ ch /\ th)))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   /\ w3a 934   = wceq 1642   e. wcel 1710  A.wral 2615   u. cun 3208  {csn 3738  {cpr 3739  {ctp 3740

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-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 2479  df-ral 2620  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-un 3215  df-sn 3742  df-pr 3743  df-tp 3744

This theorem is referenced by:  raltp  3782