Theorem rexpr 2433

Description: Convert an existential quantification over a pair to a disjunction.

Hypotheses

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

Assertion

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

Distinct variable groups:   x,A   x,B

Proof of Theorem rexpr

Step	Hyp	Ref	Expression
1		ralpr.1	. . . . 5 |- A e. V
2		ralpr.2	. . . . 5 |- B e. V
3	1, 2	ralpr 2432	. . . 4 |- (A.x e. {A, B} -. ph <-> ([A / x] -. ph /\ [B / x] -. ph))
4		sbcng 1972	. . . . . 6 |- (A e. V -> ([A / x] -. ph <-> -. [A / x]ph))
5	1, 4	ax-mp 7	. . . . 5 |- ([A / x] -. ph <-> -. [A / x]ph)
6		sbcng 1972	. . . . . 6 |- (B e. V -> ([B / x] -. ph <-> -. [B / x]ph))
7	2, 6	ax-mp 7	. . . . 5 |- ([B / x] -. ph <-> -. [B / x]ph)
8	5, 7	anbi12i 484	. . . 4 |- (([A / x] -. ph /\ [B / x] -. ph) <-> (-. [A / x]ph /\ -. [B / x]ph))
9	3, 8	bitr 173	. . 3 |- (A.x e. {A, B} -. ph <-> (-. [A / x]ph /\ -. [B / x]ph))
10	9	negbii 187	. 2 |- (-. A.x e. {A, B} -. ph <-> -. (-. [A / x]ph /\ -. [B / x]ph))
11		dfrex2 1659	. 2 |- (E.x e. {A, B}ph <-> -. A.x e. {A, B} -. ph)
12		oran 312	. 2 |- (([A / x]ph \/ [B / x]ph) <-> -. (-. [A / x]ph /\ -. [B / x]ph))
13	10, 11, 12	3bitr4 183	1 |- (E.x e. {A, B}ph <-> ([A / x]ph \/ [B / x]ph))

Syntax hints:  -. wn 2   <-> wb 146   \/ wo 222   /\ wa 223   e. wcel 960  [wsbc 1172  A.wral 1648  E.wrex 1649  Vcvv 1814  {cpr 2414

This theorem is referenced by:  r19.12sn 2448

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-rex 1653  df-v 1815  df-sbc 1945  df-un 2053  df-sn 2416  df-pr 2417

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