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  2843  ceqsex4v  2844  ceqsex8v  2846  copsexg  4329  opelopabsbALT  4346  opabm  4368  uniuni  4541  rabxp  4755  elxp3  4772  elvv  4780  elvvv  4781  rexiunxp  4863  elcnv2  4899  cnvuni  4907  coass  5246  fununi  5388  dfmpt3  5445  dfoprab2  6050  dmoprab  6084  rnoprab  6086  mpomptx  6094  resoprab  6099  ovi3  6141  ov6g  6142  oprabex3  6272  xpassen  6985  enq0enq  7614  enq0sym  7615  enq0tr  7617  ltresr  8022  axaddf  8051  axmulf  8052