Theorem rexprg 3776

Description: Convert a quantification over a pair to a disjunction. (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
rexprg	|- ((A e. V /\ B e. W) -> (E.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 rexprg

Step	Hyp	Ref	Expression
1		df-pr 3742	. . . 4 |- {A, B} = ({A} u. {B})
2	1	rexeqi 2812	. . 3 |- (E.x e. {A, B}ph <-> E.x e. ({A} u. {B})ph)
3		rexun 3443	. . 3 |- (E.x e. ({A} u. {B})ph <-> (E.x e. {A}ph \/ E.x e. {B}ph))
4	2, 3	bitri 240	. 2 |- (E.x e. {A, B}ph <-> (E.x e. {A}ph \/ E.x e. {B}ph))
5		ralprg.1	. . . . 5 |- (x = A -> (ph <-> ps))
6	5	rexsng 3766	. . . 4 |- (A e. V -> (E.x e. {A}ph <-> ps))
7	6	orbi1d 683	. . 3 |- (A e. V -> ((E.x e. {A}ph \/ E.x e. {B}ph) <-> (ps \/ E.x e. {B}ph)))
8		ralprg.2	. . . . 5 |- (x = B -> (ph <-> ch))
9	8	rexsng 3766	. . . 4 |- (B e. W -> (E.x e. {B}ph <-> ch))
10	9	orbi2d 682	. . 3 |- (B e. W -> ((ps \/ E.x e. {B}ph) <-> (ps \/ ch)))
11	7, 10	sylan9bb 680	. 2 |- ((A e. V /\ B e. W) -> ((E.x e. {A}ph \/ E.x e. {B}ph) <-> (ps \/ ch)))
12	4, 11	syl5bb 248	1 |- ((A e. V /\ B e. W) -> (E.x e. {A, B}ph <-> (ps \/ ch)))

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

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 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

This theorem is referenced by:  rextpg  3778  rexpr  3780