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	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
2exbii	⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑥∃𝑦𝜓)

Proof of Theorem 2exbii

Step	Hyp	Ref	Expression
1		exbii.1	. . 3 ⊢ (𝜑 ↔ 𝜓)
2	1	exbii 1651	. 2 ⊢ (∃𝑦𝜑 ↔ ∃𝑦𝜓)
3	2	exbii 1651	1 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑥∃𝑦𝜓)

Syntax hints:   ↔ wb 105  ∃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  4331  opelopabsbALT  4348  opabm  4370  uniuni  4543  rabxp  4758  elxp3  4775  elvv  4783  elvvv  4784  rexiunxp  4867  elcnv2  4903  cnvuni  4911  coass  5250  fununi  5392  dfmpt3  5449  dfoprab2  6060  dmoprab  6094  rnoprab  6096  mpomptx  6104  resoprab  6109  ovi3  6151  ov6g  6152  oprabex3  6283  xpassen  7002  enq0enq  7634  enq0sym  7635  enq0tr  7637  ltresr  8042  axaddf  8071  axmulf  8072