Theorem ralpr 3638

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 3634	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( A. x e. { A , B } ph <-> ( ps /\ ch ) ) )
6	1, 2, 5	mp2an 424	1 |- ( A. x e. { A , B } ph <-> ( ps /\ ch ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   A.wral 2448   _Vcvv 2730   {cpr 3584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-sbc 2956  df-un 3125  df-sn 3589  df-pr 3590

This theorem is referenced by:  fzprval  10038