Theorem ralpr 2418

Description: Convert a quantification over a pair to a conjunction.

Hypotheses

Ref	Expression
ralpr.1	|- A e. V
ralpr.2	|- B e. V

Assertion

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

Distinct variable groups:   x,A   x,B

Proof of Theorem ralpr

Step	Hyp	Ref	Expression
1		df-ral 1641	. 2 |- (A.x e. {A, B}ph <-> A.x(x e. {A, B} -> ph))
2		visset 1804	. . . . . 6 |- x e. V
3	2	elpr 2414	. . . . 5 |- (x e. {A, B} <-> (x = A \/ x = B))
4	3	imbi1i 186	. . . 4 |- ((x e. {A, B} -> ph) <-> ((x = A \/ x = B) -> ph))
5		jaob 422	. . . 4 |- (((x = A \/ x = B) -> ph) <-> ((x = A -> ph) /\ (x = B -> ph)))
6	4, 5	bitr 173	. . 3 |- ((x e. {A, B} -> ph) <-> ((x = A -> ph) /\ (x = B -> ph)))
7	6	albii 996	. 2 |- (A.x(x e. {A, B} -> ph) <-> A.x((x = A -> ph) /\ (x = B -> ph)))
8		19.26 1063	. . 3 |- (A.x((x = A -> ph) /\ (x = B -> ph)) <-> (A.x(x = A -> ph) /\ A.x(x = B -> ph)))
9		ralpr.1	. . . . 5 |- A e. V
10	9	sbc6 1947	. . . 4 |- ([A / x]ph <-> A.x(x = A -> ph))
11		ralpr.2	. . . . 5 |- B e. V
12	11	sbc6 1947	. . . 4 |- ([B / x]ph <-> A.x(x = B -> ph))
13	10, 12	anbi12i 481	. . 3 |- (([A / x]ph /\ [B / x]ph) <-> (A.x(x = A -> ph) /\ A.x(x = B -> ph)))
14	8, 13	bitr4 176	. 2 |- (A.x((x = A -> ph) /\ (x = B -> ph)) <-> ([A / x]ph /\ [B / x]ph))
15	1, 7, 14	3bitr 177	1 |- (A.x e. {A, B}ph <-> ([A / x]ph /\ [B / x]ph))

Syntax hints:   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223  A.wal 951   = wceq 953   e. wcel 955  [wsbc 1166  A.wral 1637  Vcvv 1802  {cpr 2400

This theorem is referenced by:  rexpr 2419

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-ral 1641  df-v 1803  df-sbc 1932  df-un 2040  df-sn 2402  df-pr 2403

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