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Theorem ralrimivvva 2445
Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with triple quantification.) (Contributed by Mario Carneiro, 9-Jul-2014.)
Hypothesis
Ref Expression
ralrimivvva.1  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  B  /\  z  e.  C ) )  ->  ps )
Assertion
Ref Expression
ralrimivvva  |-  ( ph  ->  A. x  e.  A  A. y  e.  B  A. z  e.  C  ps )
Distinct variable groups:    ph, x, y, z    y, A, z   
z, B
Allowed substitution hints:    ps( x, y, z)    A( x)    B( x, y)    C( x, y, z)

Proof of Theorem ralrimivvva
StepHypRef Expression
1 ralrimivvva.1 . . . . 5  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  B  /\  z  e.  C ) )  ->  ps )
213anassrs 1161 . . . 4  |-  ( ( ( ( ph  /\  x  e.  A )  /\  y  e.  B
)  /\  z  e.  C )  ->  ps )
32ralrimiva 2435 . . 3  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  B )  ->  A. z  e.  C  ps )
43ralrimiva 2435 . 2  |-  ( (
ph  /\  x  e.  A )  ->  A. y  e.  B  A. z  e.  C  ps )
54ralrimiva 2435 1  |-  ( ph  ->  A. x  e.  A  A. y  e.  B  A. z  e.  C  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    /\ w3a 920    e. wcel 1434   A.wral 2349
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-4 1441  ax-17 1460
This theorem depends on definitions:  df-bi 115  df-3an 922  df-nf 1391  df-ral 2354
This theorem is referenced by:  ispod  4067  swopolem  4068  ordwe  4326  wessep  4328  isopolem  5492  caovassg  5690  caovcang  5693  caovordig  5697  caovordg  5699  caovdig  5706  caovdirg  5709  caoftrn  5767
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