Theorem 3exbii 1538

Description: Inference adding 3 existential quantifiers to both sides of an equivalence. (Contributed by NM, 2-May-1995.)

Hypothesis

Ref	Expression
exbii.1	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
3exbii	⊢ (∃𝑥∃𝑦∃𝑧𝜑 ↔ ∃𝑥∃𝑦∃𝑧𝜓)

Proof of Theorem 3exbii

Step	Hyp	Ref	Expression
1		exbii.1	. . 3 ⊢ (𝜑 ↔ 𝜓)
2	1	exbii 1536	. 2 ⊢ (∃𝑧𝜑 ↔ ∃𝑧𝜓)
3	2	2exbii 1537	1 ⊢ (∃𝑥∃𝑦∃𝑧𝜑 ↔ ∃𝑥∃𝑦∃𝑧𝜓)

Syntax hints:   ↔ wb 103  ∃wex 1421

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-ial 1467

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  eeeanv  1849  ceqsex6v  2643  oprabid  5557  dfoprab2  5572  dftpos3  5900  xpassen  6327