Theorem ralpr 4040

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 4036	. 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 1370   e. wcel 1758   A.wral 2799   _Vcvv 3078   {cpr 3990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2804  df-v 3080  df-sbc 3295  df-un 3444  df-sn 3989  df-pr 3991

This theorem is referenced by:  fzprval  11638  xpsfrnel  14624  xpsle  14642  isdrs2  15232  pmtrsn  16148  iblcnlem1  21408  wlkntrllem2  23638  wlkntrllem3  23639  2wlklem  23642  subfacp1lem3  27237  fprb  27751  usgra2pthspth  30466  usgra2wlkspthlem1  30467  numclwwlkovf2ex  30850  ldepsnlinc  31205