Theorem 2exbii 1617

Description: Inference adding 2 existential quantifiers to both sides of an equivalence. (Contributed by NM, 16-Mar-1995.)

Hypothesis

Ref	Expression
exbii.1	|- ( ph <-> ps )

Assertion

Ref	Expression
2exbii	|- ( E. x E. y ph <-> E. x E. y ps )

Proof of Theorem 2exbii

Step	Hyp	Ref	Expression
1		exbii.1	. . 3 |- ( ph <-> ps )
2	1	exbii 1616	. 2 |- ( E. y ph <-> E. y ps )
3	2	exbii 1616	1 |- ( E. x E. y ph <-> E. x E. y ps )

Syntax hints:   <-> wb 105   E.wex 1503

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-ial 1545

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  3exbii  1618  19.42vvvv  1925  3exdistr  1927  cbvex4v  1942  ee4anv  1946  ee8anv  1947  sbel2x  2010  2eu4  2131  rexcomf  2652  reean  2659  ceqsex3v  2794  ceqsex4v  2795  ceqsex8v  2797  copsexg  4262  opelopabsbALT  4277  opabm  4298  uniuni  4469  rabxp  4681  elxp3  4698  elvv  4706  elvvv  4707  rexiunxp  4787  elcnv2  4823  cnvuni  4831  coass  5165  fununi  5303  dfmpt3  5357  dfoprab2  5943  dmoprab  5977  rnoprab  5979  mpomptx  5987  resoprab  5992  ovi3  6033  ov6g  6034  oprabex3  6154  xpassen  6856  enq0enq  7460  enq0sym  7461  enq0tr  7463  ltresr  7868  axaddf  7897  axmulf  7898