Theorem ralprg 3776

Description: Convert a quantification over a pair 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))

Assertion

Ref	Expression
ralprg	|- ((A e. V /\ B e. W) -> (A.x e. {A, B}ph <-> (ps /\ ch)))

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

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

Proof of Theorem ralprg

Step	Hyp	Ref	Expression
1		df-pr 3743	. . . 4 |- {A, B} = ({A} u. {B})
2	1	raleqi 2812	. . 3 |- (A.x e. {A, B}ph <-> A.x e. ({A} u. {B})ph)
3		ralunb 3445	. . 3 |- (A.x e. ({A} u. {B})ph <-> (A.x e. {A}ph /\ A.x e. {B}ph))
4	2, 3	bitri 240	. 2 |- (A.x e. {A, B}ph <-> (A.x e. {A}ph /\ A.x e. {B}ph))
5		ralprg.1	. . . 4 |- (x = A -> (ph <-> ps))
6	5	ralsng 3766	. . 3 |- (A e. V -> (A.x e. {A}ph <-> ps))
7		ralprg.2	. . . 4 |- (x = B -> (ph <-> ch))
8	7	ralsng 3766	. . 3 |- (B e. W -> (A.x e. {B}ph <-> ch))
9	6, 8	bi2anan9 843	. 2 |- ((A e. V /\ B e. W) -> ((A.x e. {A}ph /\ A.x e. {B}ph) <-> (ps /\ ch)))
10	4, 9	syl5bb 248	1 |- ((A e. V /\ B e. W) -> (A.x e. {A, B}ph <-> (ps /\ ch)))

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

This theorem is referenced by:  raltpg  3778  ralpr  3780  iinxprg  4044