Theorem 2exbii 1652

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 1651	. 2 |- ( E. y ph <-> E. y ps )
3	2	exbii 1651	1 |- ( E. x E. y ph <-> E. x E. y ps )

Syntax hints:   <-> wb 105   E.wex 1538

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-ial 1580

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  3exbii  1653  19.42vvvv  1960  3exdistr  1962  cbvex4v  1981  ee4anv  1985  ee8anv  1986  sbel2x  2049  2eu4  2171  rexcomf  2693  reean  2700  ceqsex3v  2844  ceqsex4v  2845  ceqsex8v  2847  copsexg  4334  opelopabsbALT  4351  opabm  4373  uniuni  4546  rabxp  4761  elxp3  4778  elvv  4786  elvvv  4787  rexiunxp  4870  elcnv2  4906  cnvuni  4914  coass  5253  fununi  5395  dfmpt3  5452  dfoprab2  6063  dmoprab  6097  rnoprab  6099  mpomptx  6107  resoprab  6112  ovi3  6154  ov6g  6155  oprabex3  6286  xpassen  7009  enq0enq  7641  enq0sym  7642  enq0tr  7644  ltresr  8049  axaddf  8078  axmulf  8079