Theorem ralpr 4086

Description: Convert a quantification over a pair to a conjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.)

Hypotheses

Ref	Expression
ralpr.1	|- A e. _V
ralpr.2	|- B e. _V
ralpr.3	|- ( x = A -> ( ph <-> ps ) )
ralpr.4	|- ( x = B -> ( ph <-> ch ) )

Assertion

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

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

Allowed substitution hint:   ph( x)

Proof of Theorem ralpr

Step	Hyp	Ref	Expression
1		ralpr.1	. 2 |- A e. _V
2		ralpr.2	. 2 |- B e. _V
3		ralpr.3	. . 3 |- ( x = A -> ( ph <-> ps ) )
4		ralpr.4	. . 3 |- ( x = B -> ( ph <-> ch ) )
5	3, 4	ralprg 4082	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( A. x e. { A , B } ph <-> ( ps /\ ch ) ) )
6	1, 2, 5	mp2an 672	1 |- ( A. x e. { A , B } ph <-> ( ps /\ ch ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1379   e. wcel 1767   A.wral 2817   _Vcvv 3118   {cpr 4035

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2822  df-v 3120  df-sbc 3337  df-un 3486  df-sn 4034  df-pr 4036

This theorem is referenced by:  fzprval  11752  xpsfrnel  14835  xpsle  14853  isdrs2  15443  pmtrsn  16417  iblcnlem1  22062  wlkntrllem2  24385  wlkntrllem3  24386  2wlklem  24389  usgra2wlkspthlem1  24442  numclwwlkovf2ex  24910  subfacp1lem3  28451  fprb  29130  usgra2pthspth  32141  ldepsnlinc  32591