Theorem 3exbii 1766

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 1764	. 2 ⊢ (∃𝑧𝜑 ↔ ∃𝑧𝜓)
3	2	2exbii 1765	1 ⊢ (∃𝑥∃𝑦∃𝑧𝜑 ↔ ∃𝑥∃𝑦∃𝑧𝜓)

Syntax hints:   ↔ wb 195  ∃wex 1695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728

This theorem depends on definitions:  df-bi 196  df-ex 1696

This theorem is referenced by:  4exdistr  1911  ceqsex6v  3221  oprabid  6576  dfoprab2  6599  dftpos3  7257  xpassen  7939  bnj916  30257  bnj917  30258  bnj983  30275  bnj996  30279  bnj1021  30288  bnj1033  30291  ellines  31429