Theorem 2rexbidv 2555

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 2531	. 2 |- ( ph -> ( E. y e. B ps <-> E. y e. B ch ) )
3	2	rexbidv 2531	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 2509

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-rex 2514

This theorem is referenced by:  f1oiso  5950  elrnmpog  6117  elrnmpo  6118  ralrnmpo  6119  rexrnmpo  6120  ovelrn  6154  eroveu  6773  genipv  7696  genpelxp  7698  genpelvl  7699  genpelvu  7700  axcnre  8068  apreap  8734  apreim  8750  aprcl  8793  aptap  8797  bezoutlemnewy  12517  bezoutlema  12520  bezoutlemb  12521  pythagtriplem19  12805  pceu  12818  pcval  12819  pczpre  12820  pcdiv  12825  4sqlem2  12912  4sqlem3  12913  4sqlem4  12915  4sqexercise2  12922  4sqlemsdc  12923  4sq  12933  znunit  14623  txuni2  14930  txbas  14932  txdis1cn  14952  elply  15408  2sqlem2  15794  2sqlem8  15802  2sqlem9  15803  upgredg  15942