Theorem 3exbii 1776

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

Hypothesis

Ref	Expression
3exbii.1	⊢ (𝜑 ↔ 𝜓)

Assertion

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

Proof of Theorem 3exbii

Step	Hyp	Ref	Expression
1		3exbii.1	. . 3 ⊢ (𝜑 ↔ 𝜓)
2	1	exbii 1774	. 2 ⊢ (∃𝑧𝜑 ↔ ∃𝑧𝜓)
3	2	2exbii 1775	1 ⊢ (∃𝑥∃𝑦∃𝑧𝜑 ↔ ∃𝑥∃𝑦∃𝑧𝜓)

Syntax hints:   ↔ wb 196  ∃wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737

This theorem depends on definitions:  df-bi 197  df-ex 1705

This theorem is referenced by:  4exdistr  1924  ceqsex6v  3248  oprabid  6677  dfoprab2  6701  dftpos3  7370  xpassen  8054  bnj916  31003  bnj917  31004  bnj983  31021  bnj996  31025  bnj1021  31034  bnj1033  31037  ellines  32259