Theorem 3exbii 1621

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 1619	. 2 ⊢ (∃𝑧𝜑 ↔ ∃𝑧𝜓)
3	2	2exbii 1620	1 ⊢ (∃𝑥∃𝑦∃𝑧𝜑 ↔ ∃𝑥∃𝑦∃𝑧𝜓)

Syntax hints:   ↔ wb 105  ∃wex 1506

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-ial 1548

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  eeeanv  1952  ceqsex6v  2808  oprabid  5954  dfoprab2  5969  dftpos3  6320  xpassen  6889