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  4330  opelopabsbALT  4347  opabm  4369  uniuni  4542  rabxp  4756  elxp3  4773  elvv  4781  elvvv  4782  rexiunxp  4864  elcnv2  4900  cnvuni  4908  coass  5247  fununi  5389  dfmpt3  5446  dfoprab2  6057  dmoprab  6091  rnoprab  6093  mpomptx  6101  resoprab  6106  ovi3  6148  ov6g  6149  oprabex3  6280  xpassen  6997  enq0enq  7626  enq0sym  7627  enq0tr  7629  ltresr  8034  axaddf  8063  axmulf  8064