Theorem 2rexbidv 2569

Description: Formula-building rule for restricted existential quantifiers (deduction form). (Contributed by NM, 28-Jan-2006.)

Hypothesis

Ref	Expression
2ralbidv.1	|- ( ph -> ( ps <-> ch ) )

Assertion

Ref	Expression
2rexbidv	|- ( ph -> ( E. x e. A E. y e. B ps <-> E. x e. A E. y e. B ch ) )

Distinct variable groups:   ph, x   ph, y

Allowed substitution hints:   ps( x, y)   ch( x, y)   A( x, y)   B( x, y)

Proof of Theorem 2rexbidv

Step	Hyp	Ref	Expression
1		2ralbidv.1	. . 3 |- ( ph -> ( ps <-> ch ) )
2	1	rexbidv 2545	. 2 |- ( ph -> ( E. y e. B ps <-> E. y e. B ch ) )
3	2	rexbidv 2545	1 |- ( ph -> ( E. x e. A E. y e. B ps <-> E. x e. A E. y e. B ch ) )

Syntax hints:   -> wi 4   <-> wb 105   E.wrex 2523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-rex 2528

This theorem is referenced by:  f1oiso  6006  elrnmpog  6175  elrnmpo  6176  ralrnmpo  6177  rexrnmpo  6178  ovelrn  6212  eroveu  6874  genipv  7841  genpelxp  7843  genpelvl  7844  genpelvu  7845  axcnre  8213  apreap  8880  apreim  8896  aprcl  8939  aptap  8943  bezoutlemnewy  12722  bezoutlema  12725  bezoutlemb  12726  pythagtriplem19  13010  pceu  13023  pcval  13024  pczpre  13025  pcdiv  13030  4sqlem2  13117  4sqlem3  13118  4sqlem4  13120  4sqexercise2  13127  4sqlemsdc  13128  4sq  13138  znunit  14938  txuni2  15252  txbas  15254  txdis1cn  15274  elply  15730  2sqlem2  16119  2sqlem8  16127  2sqlem9  16128  upgredg  16270  3dom  16903